Here are a few more stumbling blocks to be aware of:

1. #define FB_ENABLE_EXTERNAL_CLIENT must be set for Ethernet connection. For WLAN it must be deactivated. Therefore WLAN worked again after reinstalling the lib as described above.

2. you need to define the GPIO´s for Core 2 as following in the code

3. trust_anchors URL => www.google.com seems to work as Firebase is a google product

4. #include <M5Core2.h> is NOT necessary => if you use it with M5.begin you need to delete Serial.begin(115200) => otherwise your CPU will crash => known issue also thanks to felmue

5. NTP server is not working => always returns 0 => seems to work without NTP

Here is the working code i use:

/**

• Created by K. Suwatchai (Mobizt)
• Email: k_suwatchai@hotmail.com
• Github: https://github.com/mobizt/Firebase-ESP-Client
• Copyright (c) 2022 mobizt

*/

/** This example shows the basic RTDB usage with external Client.

• This example used your Arduino device and WIZnet W5500 (Ethernet) device which SSLClient https://github.com/OPEnSLab-OSU/SSLClient
• will be used as the external Client.
• This SSLClient, https://github.com/OPEnSLab-OSU/SSLClient can't use in ESP8266 device due to wdt reset error.
• Don't gorget to define this in FirebaseFS.h
• #define FB_ENABLE_EXTERNAL_CLIENT
  */

#include <Firebase_ESP_Client.h>

// Provide the token generation process info.
#include <addons/TokenHelper.h>

// Provide the RTDB payload printing info and other helper functions.
#include <addons/RTDBHelper.h>

#include <Ethernet.h>

/* 1. Install SSLClient library */
// https://github.com/OPEnSLab-OSU/SSLClient
#include <SSLClient.h>

/* 2. Create Trus anchors for the server i.e. www.google.com */
// https://github.com/OPEnSLab-OSU/SSLClient/blob/master/TrustAnchors.md
// or generate using this site https://openslab-osu.github.io/bearssl-certificate-utility/
#include "trust_anchors.h"

// For NTP time client
#include "MB_NTP.h"

// For the following credentials, see examples/Authentications/SignInAsUser/EmailPassword/EmailPassword.ino

/* 3. Define the API Key */
#define API_KEY "Key"

/* 4. Define the RTDB URL */
#define DATABASE_URL "URL" //<databaseName>.firebaseio.com or <databaseName>.<region>.firebasedatabase.app

/* 5. Define the user Email and password that alreadey registerd or added in your project */
#define USER_EMAIL "Email"
#define USER_PASSWORD "password"

/* 6. Defined the Ethernet module connection */
#define WIZNET_RESET_PIN 19 // Connect W5500 Reset pin to GPIO 19 for Core 2
#define WIZNET_CS_PIN 26 // Connect W5500 CS pin to GPIO 26 for Core 2
#define WIZNET_MISO_PIN 38 // Connect W5500 MISO pin to GPIO 38 for Core 2
#define WIZNET_MOSI_PIN 23 // Connect W5500 MOSI pin to GPIO 23 for Core 2
#define WIZNET_SCLK_PIN 18 // Connect W5500 SCLK pin to GPIO 18 for Core 2

/* 7. Define the analog GPIO pin to pull random bytes from, used in seeding the RNG for SSLClient */
const int analog_pin = 27; // Core 2 => GPIO 27 (Analog pin)

/* 8. Define MAC */
uint8_t mac[] = { 0x12, 0x12, 0x12, 0x12, 0x12, 0x12 };

/* 9. Define IP (Optional) */
IPAddress ip(192, 168, 1, 50);
IPAddress myDns(8, 8, 8, 8);

// Define Firebase Data object
FirebaseData fbdo;

FirebaseAuth auth;
FirebaseConfig config;

unsigned long sendDataPrevMillis = 0;

int count = 0;

volatile bool dataChanged = false;

EthernetClient basic_client;

SSLClient ssl_client(basic_client, TAs, (size_t)TAs_NUM, analog_pin);

// For NTP client
EthernetUDP udpClient;

MB_NTP ntpClient(&udpClient, "pool.ntp.org" /* NTP host /, 123 / NTP port /, 0 / timezone offset in seconds */);

uint32_t timestamp = 0;

void ResetEthernet() {
Serial.println("Resetting WIZnet W5500 Ethernet Board... ");
pinMode(WIZNET_RESET_PIN, OUTPUT);
digitalWrite(WIZNET_RESET_PIN, HIGH);
delay(200);
digitalWrite(WIZNET_RESET_PIN, LOW);
delay(50);
digitalWrite(WIZNET_RESET_PIN, HIGH);
delay(200);
}

void networkConnection() {

Ethernet.init(WIZNET_CS_PIN);

ResetEthernet();

Serial.println("Starting Ethernet connection...");

Ethernet.begin(mac);

unsigned long to = millis();

while (Ethernet.linkStatus() == LinkOFF || millis() - to < 2000) {
delay(100);
}

if (Ethernet.linkStatus() == LinkON) {
Serial.print("Connected with IP ");
Serial.println(Ethernet.localIP());
} else {
Serial.println("Can't connect");
}
}

// Define the callback function to handle server status acknowledgement
void networkStatusRequestCallback() {
// Set the network status

fbdo.setNetworkStatus(Ethernet.linkStatus() == LinkON);
}

// Define the callback function to handle server connection
void tcpConnectionRequestCallback(const char *host, int port) {

// You may need to set the system timestamp to use for
// auth token expiration checking.

if (timestamp == 0) {
timestamp = ntpClient.getTime(2000 /* wait 2000 ms */);
Serial.print("timestamp from NTP: ");
Serial.println(timestamp);

if (timestamp > 0) { 
  Firebase.setSystemTime(timestamp);
  Serial.print("Firebase => setSystemTime done");
}

}

Serial.print("Connecting to server via external Client... ");
if (!ssl_client.connect(host, port)) {
Serial.println("failed.");
return;
}
Serial.println("success.");
}

void setup() {

SPI.begin(WIZNET_SCLK_PIN, WIZNET_MISO_PIN, WIZNET_MOSI_PIN, -1);
Serial.begin(115200);

networkConnection();

Serial_Printf("Firebase Client v%s\n\n", FIREBASE_CLIENT_VERSION);

/* Assign the api key (required) */
config.api_key = API_KEY;

/* Assign the user sign in credentials */
auth.user.email = USER_EMAIL;
auth.user.password = USER_PASSWORD;

/* Assign the RTDB URL (required) */
config.database_url = DATABASE_URL;

/* Assign the callback function for the long running token generation task */
config.token_status_callback = tokenStatusCallback; // see addons/TokenHelper.h

/* fbdo.setExternalClient and fbdo.setExternalClientCallbacks must be called before Firebase.begin */

/* Assign the pointer to global defined external SSL Client object */
fbdo.setExternalClient(&ssl_client);

/* Assign the required callback functions */
fbdo.setExternalClientCallbacks(tcpConnectionRequestCallback, networkConnection, networkStatusRequestCallback);

Firebase.setDoubleDigits(5);

Firebase.begin(&config, &auth);
}

void loop() {

// Firebase.ready() should be called repeatedly to handle authentication tasks.

if (Firebase.ready() && (millis() - sendDataPrevMillis > 15000 || sendDataPrevMillis == 0)) {
sendDataPrevMillis = millis();

Serial_Printf("Set bool... %s\n", Firebase.RTDB.setBool(&fbdo, F("/test/bool"), count % 2 == 0) ? "ok" : fbdo.errorReason().c_str());

count++;

}
}

this Ethernet example works for me. It uses an external library for the SSL connection instead of the built in SSL client.

Thanks
Felix

After that i downgraded the library "Firebase Arduino Client Library for ESP8266 and ESP32" to 2.3.7 and after that i reinstalled 4.2.7 and it worked again. Why do i need to reinstall the lib to get it working?

/**

• Created by K. Suwatchai (Mobizt)
• Email: k_suwatchai@hotmail.com
• Github: https://github.com/mobizt/Firebase-ESP-Client
• Copyright (c) 2022 mobizt

*/

/** This example shows the basic RTDB usage with external Client.

• This example used ESP32 and WIZnet W5500 (Etherner) devices which built-in SSL Client will be used as the external Client.
• For other non-espressif (ESP32 and ESP8266) devices, this SSL Client library can be used
• https://github.com/OPEnSLab-OSU/SSLClient
• Don't gorget to define this in FirebaseFS.h
• #define FB_ENABLE_EXTERNAL_CLIENT
  */

#include <WiFi.h>
#include <M5Core2.h>
#include <Firebase_ESP_Client.h>
#include "time.h"
#include <ESP32Time.h>

//ESP32Time rtc;
ESP32Time rtc(3600); // offset in seconds GMT+1

// Provide the token generation process info.
#include <addons/TokenHelper.h>

// Provide the RTDB payload printing info and other helper functions.
#include <addons/RTDBHelper.h>

//https://github.com/arduino-libraries/Ethernet
#include <Ethernet.h>

// Include built-in SSL Client which supports other network interfaces
#include "sslclient/esp32/MB_ESP32_SSLClient.h"

// You can use MB_ESP32_SSLClient.h in your ESP32 project in the same way as normal WiFiClientSecure

// For NTP time client
#include "MB_NTP.h"

// For the following credentials, see examples/Authentications/SignInAsUser/EmailPassword/EmailPassword.ino

/* 1. Define the API Key */
#define API_KEY "KEY"

/* 2. Define the RTDB URL */
#define DATABASE_URL "URL" //<databaseName>.firebaseio.com or <databaseName>.<region>.firebasedatabase.app

/* 3. Define the user Email and password that alreadey registerd or added in your project */
#define USER_EMAIL "EMAIL"
#define USER_PASSWORD "PASSWORD"

/* 4. Defined the Ethernet module connection */
//#define WIZNET_RESET_PIN 26 // Connect W5500 Reset pin to GPIO 26 of ESP32
#define WIZNET_CS_PIN 26 // Connect W5500 CS pin to GPIO 5 of ESP32
#define WIZNET_MISO_PIN 38 // Connect W5500 MISO pin to GPIO 19 of ESP32
#define WIZNET_MOSI_PIN 23 // Connect W5500 MOSI pin to GPIO 23 of ESP32
#define WIZNET_SCLK_PIN 18 // Connect W5500 SCLK pin to GPIO 18 of ESP32

//WLAN
const char* ssid = "ssid";
const char* password = "password";

/* 5. Define MAC */
uint8_t MAC[] = { 0x12, 0x12, 0x12, 0x12, 0x12, 0x12 };

/* 6. Define IP (Optional) */
IPAddress IP(192, 168, 1, 233);
IPAddress DNS(8, 8, 8, 8);

// Define Firebase Data object
FirebaseData fbdo;

FirebaseAuth auth;
FirebaseConfig config;

unsigned long sendDataPrevMillis = 0;

int count = 0;

volatile bool dataChanged = false;

EthernetClient basic_client;
MB_ESP32_SSLClient ssl_client;

// For NTP client
EthernetUDP udpClient;

MB_NTP ntpClient(&udpClient, "ch.pool.ntp.org" /* NTP host /, 123 / NTP port /, 0 / timezone offset in seconds */);

uint32_t timestamp = 0;

const char* ntpServer = "ch.pool.ntp.org";
const long gmtOffset_sec = 3600;
const int daylightOffset_sec = 3600;
int second;
int minute;
int hour;
int day;
int month;
int year;
struct tm timeinfo;

void printLocalTime() {
if (!getLocalTime(&timeinfo)) {
Serial.println("Failed to obtain time");
M5.Lcd.print("Failed to obtain time");
return;
}
second = timeinfo.tm_sec;
minute = timeinfo.tm_min;
hour = timeinfo.tm_hour;
day = timeinfo.tm_mday;
month = timeinfo.tm_mon + 1;
year = timeinfo.tm_year + 1900;
weekday = timeinfo.tm_wday + 1;
rtc.setTime(second, minute, hour, day, month, year);
}

void wifiNtpSetup() {
M5.Lcd.setTextSize(2); //Set the font size to 2. 设置字号大小为2
M5.Lcd.printf("\nConnecting to %s", ssid);
WiFi.begin(
ssid,
password); //Connect wifi and return connection status. 连接wifi并返回连接状态
while (WiFi.status() != WL_CONNECTED) { //If the wifi connection fails. 若wifi未连接成功
delay(500); //delay 0.5s. 延迟0.5s
M5.Lcd.print(".");
}
M5.Lcd.println("\nCONNECTED!");
configTime(gmtOffset_sec, daylightOffset_sec, ntpServer); //init and get the time. 初始化并设置NTP
printLocalTime();

if (timestamp == 0) {
timestamp = ntpClient.getTime(2000 /* wait 2000 ms */);

if (timestamp > 0) {
  Serial.print("timestamp:");
  Serial.print(timestamp);
} else {
  Serial.println("NTP Server via Ethernet nicht erreichbar !");
}

}
WiFi.disconnect(true); //Disconnect wifi. 断开wifi连接
WiFi.mode(WIFI_OFF); //Set the wifi mode to off. 设置wifi模式为关闭
delay(20);
M5.Lcd.println("\nWIFI OFF!");
}

/*
void ResetEthernet()
{
Serial.println("Resetting WIZnet W5500 Ethernet Board... ");
pinMode(WIZNET_RESET_PIN, OUTPUT);
digitalWrite(WIZNET_RESET_PIN, HIGH);
delay(200);
digitalWrite(WIZNET_RESET_PIN, LOW);
delay(50);
digitalWrite(WIZNET_RESET_PIN, HIGH);
delay(200);
}
*/

void networkConnection() {

SPI.begin(WIZNET_SCLK_PIN, WIZNET_MISO_PIN, WIZNET_MOSI_PIN, -1);

Ethernet.init(WIZNET_CS_PIN);
//Serial.begin(115200);

//ResetEthernet();

Serial.println("Starting Ethernet connection...");
Ethernet.begin(MAC, IP, DNS);

unsigned long to = millis();

while (Ethernet.linkStatus() == LinkOFF || millis() - to < 2000) {
delay(100);
}

if (Ethernet.linkStatus() == LinkON) {
Serial.print("Connected with IP ");
Serial.println(Ethernet.localIP());
} else {
Serial.println("Can't connect");
}
}

// Define the callback function to handle server status acknowledgement
void networkStatusRequestCallback() {

// Set the network status
fbdo.setNetworkStatus(Ethernet.linkStatus() == LinkON);
}

// Define the callback function to handle server connection
void tcpConnectionRequestCallback(const char* host, int port) {

// You may need to set the system timestamp to use for
// auth token expiration checking.
Firebase.setSystemTime(rtc.getLocalEpoch());

Serial.print("Connecting to server via external Client... ");
if (!ssl_client.connect(host, port)) {
Serial.println("Connection failed !");
return;
}
Serial.println("Connection established !");
}

void setup() {

M5.begin();
wifiNtpSetup();
networkConnection();

Serial_Printf("Firebase Client v%s\n\n", FIREBASE_CLIENT_VERSION);

/* Assign the basic Client (Ethernet) pointer to the basic Client */
ssl_client.setClient(&basic_client);

/* Similar to WiFiClientSecure */
ssl_client.setInsecure();

config.cert.file = "/gtsr1.pem";
config.cert.file_storage = mem_storage_type_flash;

/* Assign the api key (required) */
config.api_key = API_KEY;

/* Assign the user sign in credentials */
auth.user.email = USER_EMAIL;
auth.user.password = USER_PASSWORD;

/* Assign the RTDB URL (required) */
config.database_url = DATABASE_URL;

/* Assign the callback function for the long running token generation task */
config.token_status_callback = tokenStatusCallback; // see addons/TokenHelper.h

/* fbdo.setExternalClient and fbdo.setExternalClientCallbacks must be called before Firebase.begin */

/* Assign the pointer to global defined external SSL Client object */
fbdo.setExternalClient(&ssl_client);

/* Assign the required callback functions */
fbdo.setExternalClientCallbacks(tcpConnectionRequestCallback, networkConnection, networkStatusRequestCallback);

Firebase.setDoubleDigits(5);

Firebase.begin(&config, &auth);

// Comment or pass false value when WiFi reconnection will control by your code or third party library
//Firebase.reconnectWiFi(true);
}

void loop() {

// Firebase.ready() should be called repeatedly to handle authentication tasks.

if (Firebase.ready() && (millis() - sendDataPrevMillis > 15000 || sendDataPrevMillis == 0)) {
sendDataPrevMillis = millis();

Serial_Printf("Set bool... %s\n", Firebase.RTDB.setBool(&fbdo, F("/test/bool"), count % 2 == 0) ? "ok" : fbdo.errorReason().c_str());

count++;

}
}

Excerpt from Serial Monitor:
"
Token info: type = id token (GITKit token), status = on request
[V][ESP32_SSL_Client.cpp:402] stop_tcp_connection(): Cleaning SSL connection.
Connecting to server via external Client... [V][ESP32_SSL_Client.cpp:173] connect_ssl(): Seeding the random number generator
[V][ESP32_SSL_Client.cpp:185] connect_ssl(): Setting up the SSL/TLS structure...
[I][ESP32_SSL_Client.cpp:201] connect_ssl(): WARNING: Skipping SSL Verification. INSECURE!
[V][ESP32_SSL_Client.cpp:313] connect_ssl(): Setting hostname for TLS session...
[V][ESP32_SSL_Client.cpp:334] connect_ssl(): Performing the SSL/TLS handshake...
[E][ESP32_SSL_Client.cpp:61] _esp32_ssl_handle_error(): [connect_ssl():341]: (-1) UNKNOWN ERROR CODE (0001)
[E][MB_ESP32_SSLClient.cpp:217] connect(): MB_ESP32_SSLClient Error: upgrade connection, -1
[V][ESP32_SSL_Client.cpp:402] stop_tcp_connection(): Cleaning SSL connection.
Connection failed !
"

This error indicates that there was an issue with the SSL/TLS handshake when trying to connect to an external server. The exact cause of this error is unknown, but it could be due to an incorrect SSL configuration, an expired/invalid certificate, or an issue with the network connection.
The NTP server is also not accessible via Ethernet. Therefore I connected the NTP with WLAN and synchronized timestamp with Firebase. The SSL/TLS connection still failed. I am running out of ideas.

so here my working WIFI code:

/**

• Created by K. Suwatchai (Mobizt)
• Email: k_suwatchai@hotmail.com
• Github: https://github.com/mobizt/Firebase-ESP-Client
• Copyright (c) 2022 mobizt

*/

/** This example will show how to authenticate as a user with Email and password.
*

• You need to enable Email/Password provider.
• In Firebase console, select Authentication, select Sign-in method tab,
• under the Sign-in providers list, enable Email/Password provider.
• From this example, the user will be granted to access the specific location that matches
• the user uid.
• This example will modify the database rules to set up the security rule which need to
• guard the unauthorized access with the user Email.
  */

#include <M5Core2.h>
#include <WiFi.h>
#include <Firebase_ESP_Client.h>

// Provide the token generation process info.
#include <addons/TokenHelper.h>

// Provide the RTDB payload printing info and other helper functions.
#include <addons/RTDBHelper.h>

/* 1. Define the WiFi credentials */
#define WIFI_SSID "WIFI_SSID"
#define WIFI_PASSWORD "WIFI_PASSWORD"

/** 2. Define the API key
*

• The API key (required) can be obtained since you created the project and set up
• the Authentication in Firebase console. Then you will get the API key from
• Firebase project Web API key in Project settings, on General tab should show the
• Web API Key.
• You may need to enable the Identity provider at https://console.cloud.google.com/customer-identity/providers
• Select your project, click at ENABLE IDENTITY PLATFORM button.
• The API key also available by click at the link APPLICATION SETUP DETAILS.

*/
#define API_KEY "API_KEY"

/* 3. Define the user Email and password that already registerd or added in your project */
#define USER_EMAIL "USER_EMAIL"
#define USER_PASSWORD "USER_PASSWORD"

/* 4. If work with RTDB, define the RTDB URL */
#define DATABASE_URL "DATABASE_URL" //<databaseName>.firebaseio.com or <databaseName>.<region>.firebasedatabase.app

/** 5. Define the database secret (optional)
*

• This database secret needed only for this example to modify the database rules
• If you edit the database rules yourself, this is not required.
  */
  #define DATABASE_SECRET "DATABASE_SECRET"

/* 6. Define the Firebase Data object */
FirebaseData fbdo;

/* 7. Define the FirebaseAuth data for authentication data */
FirebaseAuth auth;

/* 8. Define the FirebaseConfig data for config data */
FirebaseConfig config;

unsigned long dataMillis = 0;
int count = 0;

void setup()
{
//M5.begin();
Serial.begin(115200);

WiFi.begin(WIFI_SSID, WIFI_PASSWORD);
Serial.print("Connecting to Wi-Fi");
while (WiFi.status() != WL_CONNECTED)
{
    Serial.print(".");
    delay(300);
}
Serial.println();
Serial.print("Connected with IP: ");
Serial.println(WiFi.localIP());
Serial.println();

Serial.printf("Firebase Client v%s\n\n", FIREBASE_CLIENT_VERSION);

//config.signer.test_mode = true;
/* Assign the api key (required) */
config.api_key = API_KEY;

/* Assign the user sign in credentials */
auth.user.email = USER_EMAIL;
auth.user.password = USER_PASSWORD;

/* Assign the RTDB URL */
config.database_url = DATABASE_URL;

/* In case the certificate data was used  */
//config.cert.data = rootCACert;

// Or custom set the root certificate for each FirebaseData object
//fbdo.setCert(rootCACert);

/* Or assign the certificate file */

/** From the test as of July 2021, GlobalSign Root CA was missing from Google server
* as described above, GTS Root R1 (gsr1.pem or gsr1.der) can be used instead.
* ESP32 Arduino SDK supports PEM format only even mBedTLS supports DER format too.
* ESP8266 SDK supports both PEM and DER format certificates.
*/
//config.cert.file = "/gtsr1.pem";
//config.cert.file_storage = mem_storage_type_flash; // or mem_storage_type_sd


fbdo.setResponseSize(4096);

String base_path = "/UsersData/";

/* Assign the callback function for the long running token generation task */
config.token_status_callback = tokenStatusCallback; // see addons/TokenHelper.h

/* Initialize the library with the Firebase authen and config */
Firebase.begin(&config, &auth);

Firebase.reconnectWiFi(true);

/** Now modify the database rules (if not yet modified)
 *
 * The user, <user uid> in this case will be granted to read and write
 * at the certain location i.e. "/UsersData/<user uid>".
 *
 * If you database rules has been modified, please comment this code out.
 *
 * The character $ is to make a wildcard variable (can be any name) represents any node key
 * which located at some level in the rule structure and use as reference variable
 * in .read, .write and .validate rules
 *
 * For this case $userId represents any <user uid> node that places under UsersData node i.e.
 * /UsersData/<user uid> which <user uid> is user UID.
 *
 * Please check your the database rules to see the changes after run the below code.
 */
String var = "$userId";
String val = "($userId === auth.uid)";
Firebase.RTDB.setReadWriteRules(&fbdo, base_path, var, val, val, DATABASE_SECRET);

/** path for user data is now "/UsersData/<user uid>"
 * 
 * The id token can be accessed from Firebase.getToken().
 * 
 * The refresh token can be accessed from Firebase.getRefreshToken().
 */

}

void loop()
{

// Firebase.ready() should be called repeatedly to handle authentication tasks.

if (millis() - dataMillis > 5000 && Firebase.ready())
{
    dataMillis = millis();
    String path = "/UsersData/";
    path += auth.token.uid.c_str(); //<- user uid of current user that sign in with Emal/Password
    path += "/test/int";
    Serial.printf("Set int... %s\n", Firebase.RTDB.setInt(&fbdo, path, count++) ? "ok" : fbdo.errorReason().c_str());

    // You can use refresh token from Firebase.getRefreshToken() to sign in next time without providing Email and Password.
    // See SignInWithRefreshIDToken example.
}

}

Now i will try again to connect with Ethernet W5500.. i come back later with more infos..