I can't access my password manager (Bitwarden).

Thunderbird and Bitwarden are completely independent from each other. So you can access your Bitwarden password manager any time you want, either via the browser extension or the desktop app.

Having said that, Thunderbird doesn't support passkeys, so use password login instead.

I was assuming that login window is my browser Firefox, just with all the features turned off. Are you saying it's actually Thunderbird's own thing, thus have no way to access my browser plugin?

Normally, to log in using passkey (not password), Google calls my browser plugin directly, a confirmation from Bitwarden shows up for me to click on. With Thunderbird's pop-up window, Google does not seem to be able to request passkey from Bitwarden. The login window does not show toolbar nor any of the browser plugin, and right click does not bring up context menu, so I can't access my password manager from there.

Are you saying it's actually Thunderbird's own thing, thus have no way to access my browser plugin?

Yes, that's what I'm saying. This is a Thunderbird window. And Thunderbird and Firefox have got nothing to do with each other. So you'd have to use password authentication with Thunderbird.