Apple digital rights management (DRM, FairPlay in this case) is very complex, particularly when one adds Family Sharing or has an atypical Apple ID setup.
In our case, for reasons that made sense 10-15y ago, my iCloud Apple ID is different from my Media & Purchases Apple ID. My iCloud Apple ID is the family organizer and my Media Apple ID is a family member.
Over the past few years I've been trying to migrate to using a single Apple ID on my phone. I have migrated all but one family member.
Migration has been difficult. I don't think Apple has published a transition guide. You can't, of course, transfer purchases or media or subscriptions. There's a risk of losing a lot purchases and Apple is unlikely to help.
I looked at doing a test migration on a macOS Monterey account of mine but it seemed Monterey did not a user to change only their Media Apple ID. [Later
I found you can. In Monterey, unlike iOS, it's obscure how you do this; it doesn't show up in an Apple ID. You change the Media Apple ID through the App Store (Sign Out, Sign In).]
Since it seemed couldn't test on Monterey without trying a full Apple ID transition I made the changes on my iPhone. Let's say my iCloud Apple ID was "Sam" and my Media Apple ID was "Linda". So my device Apple ID configuration was Sam/Linda. After the change it was Sam/Sam. Sam is the Family Organizer, Linda is a family member. Linda owns our app and media purchases -- at least that's where they show up when I look.
I was particularly curious how Apple Music would work including test playlist sharing. Unfortunately I couldn't test the playlist sharing because Apple Music didn't work at all! As far as iOS was concerned I didn't have an Apple Music subscription. It offered to give me 6 months free. I also didn't have any Playlists or other configuration. Music (iTunes) configuration is tied to the Media Apple ID, not the iCloud Apple ID.
To recap, Sam is family organizer and Sam/Linda purchased the family plan Apple Music subscription. Linda shows up in Family Sharing as a family member. Once I became Sam/Sam I had no access to Apple Music. Reviewing Family Sharing it appeared that Sam should have access to Linda's Apple Music subscription. That doesn't work.
... Make sure that you're using the same Apple ID for Family Sharing and Media & Purchases...
They don't say how to migrate to that idea of course! Obviously it was possible to use a different Apple ID for Family Sharing and Media (Apple Music worked before). I don't know if the changes made to my device impacted any other family members (wish I'd checked!), but it appears for a Family Organizer device to see Apple Music they have to use the same Apple ID used at time of purchase.
Somewhat surprisingly Apple let me revert back to Sam/Linda on my iPhone. (I think there was some time limit/change limit on Apple ID media changes.) After a period of sync I had my old playlist and Apple Music access.
My guess is that to make the change to Sam/Sam and keep Apple Music I'll have to end my current subscription (tied to Linda) then change the Media Apple ID then resubscribe for the family. (In practice I'll end all subscriptions for Linda before the change.)