I have got exactly the same issue. Have you figure it out?
Microsoft authenticator ios app something went wrong when adding personal account

Hi,
I can't seem to add my personal account to the Microsoft Authenticator app on IOS. I have the latest version of the OS and the app on my iPhone but whenever I add my account, I get the message 'something went wrong'.
I'm able to get all the way to the screen where I enter my personal email ID and password. When it goes back to the screen with the 3 options where you can add a 'personal account', 'work or school account', or 'another account type'; after a few seconds I see the error message of 'something went wrong'.
Also, I've noticed after that initial attempt to add my account, that when I tap on 'add personal account' a new pop-up appears labelled 'choose one' after the pop-up with the options 'sign in with microsoft' and 'scan qr code'.