Can't move Microsoft Authenticator to new Apple iPhone

Question

We (a colleague of mine and I) need some advice about how to move the app from one iPhone to another.

As suggested elsewhere the user has made a private Outlook email account and activated it as the backup account along with an iCloud account.

It's the exact same backup information we can see on both devices but none of the 4 accounts setup in the Authenticator app on the old phone have been moved or syncronized to the new phone.

Isn't that exactly what it should be doing when you have selected the setup we have (with iCloud and a personal Microsoft account)?

Or have I missed something?