I don't know about an "authenticator app". I don't recall having to involve anything like that and my Gmail account works as expected. You don't mention what version of Thunderbird you're using, but recent versions have a new feature called "Account Hub" which is classified as "experimental". Some people have had problems with it, though it's worked well for me. I have a feeling it works best with common email services like GMail, less so with more obscure ones. You might try turning it off (see Tools | Settings | General) and see if that helps.

Check out Google's troubleshooting suggestions if you haven't already.

https://support.google.com/mail/answer/7126229

I was able to solve my problem by completely removing my google account and all connected data relating to it, then reinstalling it using the two-factor authentication for my account installation.

I have this same problem, except I have already tried the majority of online solutions including purging cookies and passwords, deleting/re-adding my account, clean uninstall/reinstall of Thunderbird, but it still fails to authenticate. Sometimes it even just loops on the two-factor authentication prompt. The closest I've ever gotten to getting it to work was using an App Password instead of OAuth2, but at best it only allows me to receive e-mails but I cannot send them.

Also it was working until somewhere around late October. I have never messed with Thunderbird settings before then.