At this point, I would stop treating this as a normal Apple ID or password problem.
Since you can sign in to your Apple Account online, the App Store and System Preferences show you as signed in, Apple’s System Status is green, and you have already performed a completely fresh Catalina installation, the problem appears to be specifically with Apple’s media authorization on Catalina.
The strongest clue is that Music and Apple TV both give you the same generic error when you try to authorize the iMac.
Apple confirms that authorization is what allows a Mac to access purchases, and that once a Mac is authorized in one Apple media app, its purchases should be available in the other apps as well. (Apple Support)
Apple's own current guidance for problems connecting to Apple Music, Apple TV, the App Store and related services recommends checking System Status, verifying the network, ensuring the date and time are correct, trying another network, and updating macOS. (Apple Support) You have already eliminated most of those possibilities.
I therefore think it is quite possible that Catalina's older Apple media components are no longer communicating correctly with Apple's current authentication/authorization servers.
That does not necessarily mean Apple has officially declared Catalina "dead"; it could simply mean that something changed on Apple's server side and the older software is no longer handling the authentication process correctly.
A fresh installation would not fix that because it installs the same Catalina components.
The most useful test now would be to try the same Apple Account on another Mac running a newer macOS.
If that Mac can authorize successfully while your Catalina iMac cannot, that would provide very strong evidence that the problem is Catalina rather than your Apple Account.
You could also try the iMac temporarily through a completely different Internet connection, such as a phone hotspot. Apple specifically recommends trying another network for problems connecting to its services. (Apple Support)
I would also not create a new Apple Account. Your purchases are associated with the account with which they were originally purchased.
Apple specifically states that an authorized computer is required to access previous iTunes Store purchases. (Apple Support)
These are the Apple pages I would keep handy:
Apple: Authorize or deauthorize your computer for Apple Account purchases
Apple: If you can't connect to the App Store, iTunes Store, or other Apple services
Apple: Redownload music from the iTunes Store on your Mac
Apple: Download previous iTunes Store purchases in Music on Mac
So, based on everything you have already done.
I would not keep reinstalling Catalina or repeating the standard Apple Support troubleshooting.
The authorization failure itself is the key issue, and testing the account on a newer Mac would be the cleanest way to establish whether Catalina is the culprit.