OK, I found the reason.
I am using the AdobeID I use for CCP also on my local machine to use CC programs
As I currently install a new workstation, I have this ID on two machines.
So, both "slots" for this account were taken and CCP could not use this AdobeID anymore.
Unfortunately, it does not tell you about this, and also it does not give you any options to use a different AdobeID or to log out the other IDs. It just tells you "Something went wrong" [insert smiley for 'stupid' here]
The only way to fix this was, to install a local CC program and start it. Then it tells you, that you have no free slot left. Even though, I unregistered my old machine already, it did not work until I told it to logoff any assigned machines.
[insert another smiley for 'even more stupid' here]