Okay, after some further tests it seems, that the Authenticator only fails on my Pixel 3a.
It works on another Pixel 3a (Android 10) and on a Pixel 2 XL (Android 10). So, Android 10 is definitely not the culprit.
The question is: What keeps the App from working on my Pixel 3a?
A few more hints: I had the app installed and it worked. I tested it with some TOTP-Tokens and since I did not see any advantage over other authenticators I uninstalled it. Then, after starting to test PUSH-Tokens I reinstalled it because these won’t work with other apps. So only the ‘second instance’ does not work.