I've been watching other firebase related repositories. And I think I got an answer (solution) for this issue:

To update the phone number that is used as second factor, proceed as follows: login the user again with MFA (e.g. use signInWithEmailAndPassword for email as first factor, followed by verifyPhoneNumber for the second factor) when the MFA login successfully completed, enroll the new phone number (again using verifyPhoneNumber, but with the new phonenumber as input parameter) when the new phone number is successfully added, unenroll the original phone number (using the MultiFactor.unenroll -> https://pub.dev/documentation/firebase_auth/latest/firebase_auth/MultiFactor/unenroll.html) More details on step 1) and 2) can be found at https://firebase.google.com/docs/auth/flutter/multi-factor