I started to have this issue two days ago on my trusted Iphone SE 1gen (iOS 15.8) and Ipad Air (iPadOS 17.2). Constant requests to enter the "right" outlook.com password, but when I do it's not accepted, and the mail app is not updating with new emails. Both phone and iPad have outlook.com on the Apple mail app as Exchange accounts, which is what Microsoft recommends: https://support.microsoft.com/en-us/office/set-up-an-outlook-account-on-the-ios-mail-app-7e5b180f-bc8f-45cc-8da1-5cefc1e633d1
This new behaviour was not connected with me doing any recent iOS or iPadOS updates. My Macbook that also has outlook.com on apple mail app as Exchange account did not have any problems.
On my Ipad I tried to just turn off "Mail" in the account settings, shut down iPad and reboot, turned on "Mail" again, and this fixed the issue so far.
Did exctly the same on the phone, but it did not help. Then I deleted the mail account, turned off the phone and rebooted, and added the Exchange account again, but that did not fix it either. Tried a couple of more times to just turn off and on "Mail" in the settings, did not help.
Finally I deleted the Exchange account and added an Outlook.com account instead, and that seems to have fixed it on the phone.
Does anyone know if there is any drawbacks on having the account as an Outlook.com account instead of an Exchange account?
(Sorry about double posting, this first ended up in the middle of the thread which was not my intension, the previous post from me can be deleted)