It's not fixed.
I had it on my old iphone and authenticator worked fine. I got a new phone, so I need to log into authenticator... by using authenticator.
I'm stuck in this loop and I've tried multiple devices. This is so dumb!
Every post in this thread that mentions a solution that works says exactly the same thing.
To solve the problem you need to log into the account using another method other than the authenticator (password or Windows Hello (biometric), or text/email a code, etc.), then delete the authenticator option titled 'Send sign-in notification' from the list of verification methods in the account security section.
Once you've removed the old, invalid authenticator selection from your account the problem goes away, because it's no longer there to be used and you can set up the new phone's authenticator.
The reason it happens is simple, you got a new phone and didn't take the time to find out first how to properly switch from one phone to another or couldn't because you lost the old phone.
For an individual Microsoft Personal account user, the process to fix is as I described above.
For anyone with either a Work or School account, the answer is similar, but varies depending on the way their account is managed. The authenticator, possibly titled; MFA option, must be deleted or disabled. Once this is done it will allow setting up a new authenticator to take its place. The problem with Work or School is really who needs to do this, since typically it's the administrator of the organization, who may be an IT department, but for small firms may be the same person who owns the account, which means they might need Microsoft Support's help if they can't get into the account without the old authenticator to fix it.
I won't try to explain the Work or School possibilities here, since the answer is 'it depends'. There's simply too many possible people and/or management interfaces involved and no one but you could possibly have that information, so it's pointless to try and answer in a single post. My only recommendation is simple, see your Administrator. If you're the Admin, contact Microsoft Support.
Rob