Catalina Mail on new MacBook Pro rejecting valid passwords

It might help to know more about the setting here: who is the email provider, are these POP or IMAP or Exchange emails, etc.

How did you transfer the email accounts from the old to the new computer? Did you use Migration Assistant? I have never seen this problem using Migration Assistant.

One thing to try: in Mail, go to one of the problem accounts, open Preferences and from within Mail Preferences, select Accounts. Go to the problem account and check all the settings, every single one, and re-enter the correct username, email address, and password (if those are presented; for some email like Gmail some of those items won't be entered there).

Then close Mail and go to System Preferences, Internet Accounts, and select the same problem account. Re-enter the information there and also go to Advanced button and re-enter everything there. You might think that these are redundant items and should mirror each other, but I have found with certain email providers that the info needs to be re-entered in each place.

If you did not use Migration Assistant to transfer from old to new computer, then how did you do the transfer?

The "Unable to verify account name or password" message is decidedly unhelpful. The only meaningful inference is that Mail failed to authenticate for any number of reasons.

First, please read If Mail on your Mac keeps asking for your password which specifically addresses the perennially problematic Google Mail.

Then, quoting from that document:

"If no other solutions work, try setting up your email account again. To do that, first remove your email account, then add the account back."

Wow, that's a lot of email accounts!

With Exchange and GMAIL, are you using them in an IMAP-like way, where your Mac basically mirrors what is on the servers, and all your emails with those accounts are on those servers?

If so, one thing you can consider doing (maybe do a complete up to date backup first) for those not working accounts is to delete the account from your Apple Mail AND from System Preference Internet Accounts. Maybe reboot into safe mode just to empty caches and anything that might "remember" them. The add them in as if they were "new" email accounts. The email on the servers will download immediately and populate your Inbox, etc.

If you have some locally (On My Mac) stored folders with emails from those accounts, export or save those separately before deleting the account as they will be deleted along with everything else. Best way is to export them in such a way that they can be re-imported once the email account is working again.

One last thing -- can you properly log in and access emails via webmail for those not working accounts?

Shouldn't be this hard. I like Apple Mail (despite its primitive search capability) but the password business in Apple Mail can be confounding. I have an Exchange account for work, and the password must be changed every 3 months. Under Outlook on the work computer, it's trivial, one preference. Then I run Apple Mail on my home computer and it seems I have to try multiple times, both from within Mail and in the System Preferences, before it works. Maybe I have anticipatory tenseness and mistype it or maybe it's something else.

Another thing which I was going to go through with you but luckily you did not have to ... the Keychain sometimes incorrectly keeps multiple entries for mail passwords. Then it can sometimes read the wrong or older one and the password does not work. This happened to my wife, about every two weeks Mail would request a valid password. I finally located the older (incorrect) Keychain entry and deleted it, and the problem was resolved. I think you ran into a version of this when your older computer might have been ignoring an older or incorrect duplicate Keychain entry but after Migration the new one couldn't tell which one to use. Since everything is now working I would let it lie ... but if you encounter it again, that might be a place to look. Two entries in Keychain, one with an older or incorrect password.