password verification error

The most expedient way to fix that problem is to remove the problematic email account(s), and add them again.

The "unable to verify account name or password" is the best Mail will offer. All that can be inferred from that message is that it was unable to verify those email account credentials — it does not mean they were incorrect.

All of those are possible. For example it is no longer possible to use "old" versions of the Mac Mail app to access email messages hosted on the latest versions of the Microsoft Exchange server.

Exactly how old is "too old" is a function of the particular Exchange server software. You may have been using Mail (or MS Outlook for that matter) to access those messages just fine, until one day you can't. That's an indication the email hosting service updated their software. Needless to say if your Mac is running a very old macOS version — on the order of fifteen years or so — then you won't be able to use Mail for those MS Exchange-hosted messages. Removing the problematic email account and adding it again will fail: you'll get the same unhelpful "unable to verify account name or password" message because it cannot establish the secure email message protocol required, and you're stuck. You probably won't even be able to install the latest version of the Microsoft Outlook software. You may however be able to use a web-based Microsoft email client... unless of course your Mac can't run the latest web browser...

Anyway that's one reason the possibilities you mention are all on the table.