Slightly different but similar issue. I've had Apple stuff from very early before there was a .me or .icloud and thus my original Apple ID was another email. When I bought iPhone, iPads, I had created a .me ID. But purchases were still on the other ID. I think there were a bunch of us from the early days like that. Recently I bought a new iPad (first device in a couple of years). Bought it using my .me ID. Then I received and email about 1 year free Apple TV+. On my Mac Book .me is only ID. Others devices list the .me as the ID but purchases under other. When I applied the free year it only shows up on my Mac Book with everything including purchases set up as .me. Can't get it on my other devices with different purchase ID. Talked to Apple and I get there is "nothing" they can do about it.
I've thought about getting rid of the other ID but afraid I'd lose all my purchases. Oh well probably won't watch much TV+