Unable to activate iMessage on macOS Catalina

I am unable to activate iMessage on macOS Catalina, whereas iMessage works for my AppleID on iPad and an older macOS version.

Also, another AppleID can be used to activate iMessage on Catalina on my Mac computer.

So, it seems the problem is specifically related to my AppleID in combination with my Mac computer with Catalina installed.

I re-installed Catalina, changed my password, enabled two-factor authentication, disabled and cleaned up Keychain, etc. as suggested by several questions on this forum. None of these solutions works for me.

I also created a second AppleID and I do have the exact same problem with this new AppleID. So, some AppleID's work (e.g. my parents'), some don't (my one and my second one).

Any suggestions?