Yes its possible. It's in fact the way Family Sharing and the Apple Music Family Plan is intended to work. Each person has their own library.
If you are seeing everybody's playlists and library, then you are doing something wrong. It suggests you are using the same Apple ID on everybody's devices which essentially renders Family Sharing useless.
Purchases and subscriptions such as Apple Music, are tied to the Apple ID that buys or starts them. As such every device logged into said Apple ID has access to the content and subscriptions from that Apple ID, so Family Sharing is not required nor able to be used.
To do what you want, each person needs to have their own Apple ID logged into their devices, for iCloud and iTunes & App Stores.
Then one person, as the Family Organizer, creates a family group, and invites everybody else to join using their own Apple ID.
That way, everybody's account is separate and as such each person gets their own music library and does not see everybody else's music and playlists unless explicitly shared by them.
See here: Set up Family Sharing - Apple Support
here: Get an Apple Music family subscription - Apple Support
and here: Family Sharing - Apple Support