I'm just gonna go ahead and say that I'm not sure what happened there but either you or your mom signed in with your account on the other device.

I have a lot of technical understanding with how CloudKit works and there's not a pathway for what you're describing to come out of a family group.

Maybe Something to do with Family Purchase Sharing. I didn’t realize when I bought an audio book it would appear in my dad’s library. Kind of embarrassing. Apple’s help pages make it sound very opt in but I think there are bugs where libraries are merged by default. Some say on a quiet night you can still hear Bono singing “sexy boots”…

Libraries are not merged, only purchase history. It does not download to their device in any scenario automatically.

A lot of people have their iTunes accounts signed in on other devices which would do what you describe, but not family sharing.

Hence, "buggy".