I think I need to clarify a couple of points that I think some of you are having with this issue and with my solution.
You have no need to change your Apple ID, because all of your music is linked to that.
The problem relates solely to the device user account in
System Preferences > Users & Groups.
I found that when I had this issue with Apple Music when logged-in on my iMac; that my wife could sign-in to her Apple Music account on my iMac even though she was not logged-in to the iMac. This indicated to me that the problem was not related to Catalina 10.15.1.
It indicated to me that the problem was related to my Log-in on the device.
By adding a new Admin Account using my Nick-Name + Surname (instead of First-Name + Surname) and using my same Apple ID on this new account, the issue was gone.
Once you have made another User/Admin Account to log-in to the device/machine; de-authorise Apple Music and then log-out of the device on the old User/Admin Account.
Then log-in to the device/machine using your new User/Admin Account. Open Apple Music and re-authorise the device/machine. Close Apple Music and re-open it and you should see in the bottom left-hand that it is updating artwork. Let it complete this. When it has finished then you will be able to sign-in to Apple Music and you are set to go. Once Apple Music is working then you can get rid of your old User/Admin Account by going back to System Preferences > Users & Groups and deleting the old User/Admin Account.
To me it doesn’t matter that I am using ‘Fred Bloggs’ instead of ‘Frederick Bloggs’ when I log-in to my device/machine as all my data is still there under my same Apple ID.
Hope this helps!