I have it resolved but wanted to share, for others to potentially benefit.
I had Apple Chat recommending to reinstall macOS. One can do this through clicking (keep it clicked!) the command key+R on the Mac while booting. Then after a while a menu pops up with one option to reinstall. This will take about an hour (in my case).
Did not change the problem however in any way.
So I called Apple Support and had a senior supporter eventually on the phone.
He really took the time to resolve it.
In the end it was probably a corrupted file which was attached to my user account and therefore got not removed when reinstalling macOS. So we created a new user during the boot action. With this user we removed the user and with it obviously the corrupt file. There was a complicatioon since the corrupt user was the login user when the system booted automatically. Therefore we had to also reset this, so that this user would nt be automatically the user when booting. Only then was the user deletable.
In addition creating the new user one can assign the Home folder of the other user, so that everything is preserved.
After the new user with the Home folder of the corrupt user was created and the corrupt user removed, everything was working again, nothing was lost.
Thanks to Apple support for so efficiently helping.
