Hey, if we’re lucky, maybe I can give vague guidance that could actually help somebody out! Please, don’t shoot me if it doesn't. But I actually got out of the loop successfully and will here try to share my experience by my best ability. It requires access to the Microsoft 365 admin center, so if you are not an administrator you will first need to go to her or him. Sorry if everybody in your organisation is locked out, then this won't save you. In my case, the accounts are workplace type accounts.
• Have the administrator log into the Microsoft 365 admin center.
• In the left menu, find Users -> Active users
• Above the user list itself, a menu bar emerges with entries like Add a user, User templates ... Click “Multi-factor authentication” (MFA).
• This gives a new tab, new user list, and MFA status per user in one column. (In my case, funnily, my looping colleague showed status “inactive” – so why was he forced to use Microsoft Authenticator if MFA was even inactive?!)
• Mark the problematic user in the list, and some options appear on the right. Click “Manage user settings” (or equivalent, I am on an exotic language myself so can’t tell literally).
• A dialogue opens with three checkboxes. And here goes the crazy thing: Tick “Require selected users to re-enter contact methods”, then “save”. (Again, not literal but by my spontaneous translation.)
• At this point, I don’t fully remember, but somehow it generated a new user experience for the looped user. Starting on the laptop, it was possible to enter the account, with access to some user settings, including some security settings. Fiddling around, and clicking “Next” on some prompt, there was a QR code appearing for the possibility of connecting the Authenticator app by scanning it.
• I think we deleted the account from the phone Authenticator and launched it again from scratch. I guess we selected the workplace account type, then the QR reader method.
(By the way, I am thinking it might be a positive thing to populate the account with a mobile number, a backup email address etc, to enable alternatives to the Authenticator until it works. This is what I tried to do, at least.)
This solution happened only after loads of lost hours of trying this and that, and the solution is the least intuitive you could imagine so it’s not strange there are currently 420 me-toos on this post and seemingly no solution presented by MS (which, if so, tells you something about MS, too, I guess). At my help was Vivek G. on Freelancer, so cred to him for providing the key idea in figuring out.
Hope this could work for you folks out there, too!