I have 3 Microsoft Office 365 tenants on my computer, along with one personal free Outlook account. Some accounts use multi-factor authentication (MFA) and others do not.
One symptom of the problem I have had is not being able to put in a password in local Outlook. This problem has persisted across different versions of Outlook. The work-around I found was to never save my password in Outlook. This typically worked, but not all the time. Also, one time accidentally hitting the prompt to save the password on an account with MFA guaranteed I would have this issue again in about a month.
I started having an issue with OneDrive where I could not login to one tenant. It would let me put in the username/password for one tenant, then give me an error saying the user did not exist in the tenant, with the error showing a different tenant than the one I logged into.
The fix for OneDrive login authentication wound up being simple and it fixed both issues. Here are the steps:
1) Close all local office applications. This includes logging out of OneDrive and OneDrive for Business. (Show hidden icons on your taskbar to get access to the OneDrive cloud icon, then quit OneDrive.)
2) Delete these two folders:
After that I launched Outlook and it immediately connected and updated the tenant accounts that previously refused to connect. I also launched OneDrive and all of my tenants authenticated properly, connected and synced my files.