Turingtest2, your comment was valuable! Carefully examining the iOS screen, my son's name is attached to the playlist, not the song. As an experiment, I created a new playlist on my Mac within Music, and then synced to my iPhone. The new playlist is there, and under its title appears my son's name, not mine, and "updated today".
So I agree with you that it appears my son is not seeing my private library. Instead iOS is confusing my identity and my son's identity within the Music app.
In Contacts iOS app, my own entry is noted as "my card" and my son's card is not shown as "my card" -- however all my children are not only logged into iCloud under their own AppleID, but they are also logged in under my AppleID. iOS allows you to log in under multiple AppleIDs.
Why do we do this? It is a vestige of the days before Family Sharing. In order to share apps, calendars and purchases, everyone used a single AppleID. Although that is no longer required, I think my family would lose a bunch of apps and music (or at least be required to re-download them all as a family sharing purchase) if they logged out of my AppleID.
The Music app is confused on the identity.