On an iPad 6th generation, I upgraded to 15.2 on 14 Dec 2021 began to observe the problem on the evening of 15 Dec.
(I have not observed the problem for the same outlook accounts on an iPhone 7 also running ios15.2)
The problem occurred for both outlook accounts in my accounts list.
I was repeatedly prompted for both passwords.
I would re enter the password, but nothing seemed to happen when I hit the submit button.
when I looked at the account, I could see that it was still requesting I re enter the password.
I restarted the iPad and still observed the same behavior.
I deleted one of the outlook accounts but could not add it again. Today, the one remaining outlook account seems fine.
I attempted deleting and re-adding one of the outlook accounts, but cannot re-add the account.
1) I tested my username/password combination by logging in via the outlook.com web interface.
2) I used “add account”, typed in the username and password when prompted, then used the “Submit” button
3) I observed that the process stopped at that point - I was looking at the list of accounts. The new account was not added.
There were no error messages or any other type of message after I used the “submit” button.