Me and the wife are having the same issue.
I use BlackBerry Hub on Android, started prompting at the start of the week. Managed to get it going using different server URLs for a short time, e.g. outlook.com, outlook.office365.com, hotmail.com, microsoft.com with varying degrees of success. Outlook.live.com seemed to work for a few hours then got the same error.
Wife just uses native iOS mail client and is also prompted for password.
I also experienced a period of time (yesterday?) where the issue seemed to have abated but it had not and returned as bad as before.
Utter shambles, considering moving away from Microsoft for mail. Wasn't keen on Google but at least I wasn't stuck without mail on my phone for a week.
Where misery loves company you may appreciate this. Recently "cut the cable" and committed several accounts to Outlook over Gmail. That being said, If this is just an untoward outage than I'm not so worried because those have been rare. I can still use email on the many affected devices, platforms, and configurations, (which is most of them) and The Outlook android app seems to experience no errors so that's the My last backup option, which I've not needed yet.