Moving Microsoft Authenticator to a new phone

For any account that won't restore, log in to that site on a computer while the old phone still works, go to its 2FA or security settings and set up a new authenticator. It gives you a fresh QR code to scan with the new phone. While you're in there, save the recovery codes it offers. That's what saves you next time a phone dies without warning.

Keep the old phone until this is finished. That's the main thing. On the old phone, turn on backup in Authenticator's settings (iCloud Backup on iPhone, Cloud backup on Android). It needs a personal Microsoft account. On the new phone, install the app, sign in to that same Microsoft account and choose to restore from backup. What doesn't come across cleanly: work or school accounts usually need verifying again, and some other accounts may need re-adding. Go through the list and check each one works before you wipe the old phone. Also, the backup won't cross between Android and iPhone, so if you're switching platforms, every account has to be re-added by hand.