Can each family member in the music family plan have their own music and playlist on their phone ?

If this follows a common path for these discussions, you’re sharing an Apple ID somewhere.

I’m unclear about what is getting synced, too. There’s no need to sync a library with a Mac or PC with Apple Music, as that sync happens automatically when iCloud Music is enabled.

Settings > tap your name > iCloud > Show All > Music ✅

Which all makes me wonder if it’s the sharing of the Apple ID associated with a Mac or PC system used to sync.

I’d also look for a local Apple Music subscription, and which Apple ID that’s using. (I’d think Apple Music would flag that and offer to cancel that local subscription, though.)

The iCloud Music option that enables the music syncing is labeled as "Sync Library" on iOS devices and Macs.

Use Sync Library with your Apple Music subscription - Apple Support

There may be a confusion about what it does though. It will not sync to others libraries, only to libraries on the devices logged in to the same Apple ID.

Which as you point out, means somewhere someone used someone else's Apple ID and merged the library and playlists at some point. Or there is a PC somewhere that has a merged library.

Otherwise, the merging of libraries and playlists should not be happening.

JoshChesshir wrote:

We do not want to see the music everybody downloads and see their playlist. We’re signed in on our own accounts with Apple but when we go to start a playlist it wants to sync the library and then we see all the music everybody has downloaded.

Then it sounds like you did something wrong.

If you are in fact signed in with your own Apple IDs under Settings ➜ Apple ID ➜ Media & Purchases on your own devices then each person's Music Library and Playlists are entirely independent.

There's no way for you to see other people's library when using Family Sharing correctly.

Are you sure you are correctly signed in?

Are you sure nobody else is using a common Apple ID on their devices?

"Sync Library" syncs the library from the specific person's devices, devices signed in to their Apple account only. It should not be syncing everybody else's library or playlists at all unless again somewhere some device was logged in to somebody else's Apple ID and the libraries merged there for some reason.

Libraries are not mixed or merged in Family Sharing at all unless something was done incorrectly.