You can integrate Firebase Authentication with a custom authentication system by modifying your authentication server to produce custom signed tokens when a user successfully signs in. Your app receives this token and uses it to authenticate with Firebase.
While the API provides multiple methods for authentication, we strongly recommend using OAuth for production applications. The other methods provided are intended to be used for scripts or testing (i.e., cases where full OAuth would be overkill). Third party applications that rely on GitHub for authentication should not ask for or collect GitHub credentials. Instead, they should use the OAuth web flow.