Since Catalina: not able to log in to Apple Music, iTunes store, TV+, Podcasts

After my upgrade to Catalina, everything was first running smooth (except Time Machine, see my other post). Just the iCloud panel in the system preferences complained that I would need to log in again in order for some services to function correctly. I did not see which service wouldn't work, as everything (incl. Music, iTunes, TV+, iCloud drive, ...) worked perfectly. But to get rid of the annoying message, I did chose to log out of my AppleID and then in again.

Turns out this was a bad idea. The warning message in the preference panel went away at least, so mission achieved in that respect. Just: since then, I was thrown out of Apple Music, TV+, iTunes store, ... Being "thrown out" in the sense that it now simply shows the promotional panel asking if I want to try our Apple Music. If I click on "log in" in that panel, or chose "log in" in the "Account" menu, I have to enter my AppleID and password. While I do not get an error message, I am also not logged in, though. Nothing changes! Clicking on "iTunes store" sometimes gives an error ("check you network connection"), sometimes iTunes store loads up normally, just not having me logged in. I tried on a different user account (same machine, same AppleID) and it works! I re-installed Catalina (over the existing installation again), but the problem with my main user remains :-(

I know I could do a clean install, but honestly, it will take hours to re-install all relevant applications and user data (email, OMG), so I am VERY hesitant to do so and might rather wait for the next OS update.

Or is there ANYBODY out there who could help me?