Since the original owner of this thread never selected an answer and it's grown into a mess, I'm modifying my initial post to include a general answer for new readers. It doesn't answer all possible questions as you'll see but does indicate the general problem and solution for Microsoft Personal account users.
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 option, possibly titled; MFA, must either 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 account owner himself, 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. If you prefer, read through the other posts in this thread to see if someone else's situation sounds similar to yours.
Rob
All text below this point was from my initial post before we understood the above, which I'm leaving only to provide the history that supports some of the posts that follow. There's no reason to read it for most, since it's didn't truly fix the problem.
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Do you have a biometric login option set on your phone like your image or fingerprint reader?
With that method enabled on my phone, I'm always prompted for my fingerprint each time I reopen the app, even if I just minimize it, but I can move between selections on the bottom bar without a new challenge while it's open. Been a couple years since I set mine up, but don't ever recall seeing an option to sign-in displayed.
Also in Settings (3 vertical dots at top right of app) is a selection under Security called Screen Capture that allows apps to do that, I assume that would work for your purpose, though it's a security risk during normal operation.
Don't know how fast the new threads are indexed, but typically they'll be on the top page for the better portion of a day, and you can always click the login option near the upper right of this screen and select 'My Profile' from the drop-down that appears to see a list of the threads you've posted to, including your own, as you scroll down that page. Few ever get here by search, so your questions will typically be answered by volunteers like me or contract workers with a Microsoft tag, so visibility via search is unimportant anyway.
Rob
< EDIT > Now that I think about it, you don't have your initial Microsoft account setup in Authenticator yet, which is why you're getting that prompt. If you're setting up a personal account like mine, here's the simple page for one of those.
How to use the Microsoft Authenticator app - Microsoft Support