Outlook 2016 doesn't accept password or app password

Your observation is very consistent with mine. It seems in the last couple of days, the prompt happens when I switching from mobile to non-mobile data network, and between one network and another (happened when I enabled VPN and again when I disabled VPN). It seems like MS systems authenticates and grants permission weighing heavily on IP address, so if the device suddenly changes IP address/network, it would fail the new attempts, even if it was previously authorized on those networks. Given competing platform/email system handle the same situation gracefully, it suggest these are MS issues or possibly design flaws with their security protocols.

I've gone about 2 days now without a single disconnect on my Gmail for android with EAS, that's with multiple changes between wifi, mobile network, etc.

Anyone still having the issue, can't say whether it is fully fixed or not but you may want to make sure you have 2FA enabled and are using an app password (or modern authentication if the app supports it) and are configured to use exchange activesynch instead of IMAP (EAS is much better anyway).

Worst case may need to delete the account from the device, clear the app cache, delete the device from your MS account (mobile devices are listed under settings on outlook.com), and re-add it. A reboot between deleting and re-adding isn't a bad idea, and generate a new app password (clear old ones if you don't have any other devices using them too).

A little add from my side: Directly after leaving my WLAN Network I got a password prompt. After I got back to exact this WLAN network again (about 15 mins later, so same public IP address) I got my mails pushed again without touching the new password prompt, so it was working again.

Perhaps this helps identifying the problem.

I have not received any emails or any other updates on Nine Folders since Jan 26th. Nine is an Outlook for Android. I tried to wade through the 37 pages of responses but couldn't find the answer. Is there a fix?

You are over complicating things. One email address tie to one password, which is one and same account. We are not talking about aliases. The login email address is the account's email address. I also do not sign into the device with this account and computer's login account is a local account, so this is basically an email account being added to the email client. Same for both my work computer and personal computer. My phone is logged into a Google account for the device, so access Play Store and such. Only the email clients, GMail and Outlook for Android, are logged into these effected Microsoft accounts.

My practice is simple, I don't log into a MS account unless I have to, so my Edge browser is not logged into this MS account. I cannot imagine letting MS control even more aspects of my work and personal lives if they can't even work through this login/authentication issue.