Howdy jkid561,
It’s great to have you in the Apple Support Communities! I gather that you're wanting to determine why your girlfriend is still being charged that amount each month, and you'd like to figure out how to share the songs you see in your Sync Library that don’t appear on her iPhone. It's going to be my pleasure to help out.
If you've already sent her an invite and she's joined Family Sharing, there shouldn't be an extra charge for music.
If someone in the family group already has an individual or student subscription to Apple Music, renewal of that subscription is automatically canceled when they join the family group.
Get an Apple Music family subscription
Tap Settings > [your name] > Family Sharing. If you're using iOS 10.2 or earlier, go to Settings > iCloud > Family. Can you confirm that she accepted invitation? If so, make sure you’re sharing the Apple Music Family subscription on your iPhone by following the steps in the section "Review your Family Sharing settings" here: If you don't see your family's shared content
It might also be a good idea to have her check for any active subscriptions on her iPhone by following the steps here: If you see apple.com/bill, itunes.com/bill, or an unfamiliar charge on your statement
Apple IDs are personal accounts to access Apple services so it would be expected behavior that the music in your Sync Library doesn't appear in your girlfriend's Sync Library.
Apple Music Family Subscription
Everyone gets full access to the Apple Music catalog, with millions of songs. And each family member gets their own personal music library, and music recommendations based on what they love to listen to.
Family Sharing
If you purchase music in the iTunes Store, you can definitely share that using the steps here: Share App Store, iTunes Store, and Apple Books purchases with Family Sharing This wouldn't apply to music in the Apple Music catalog.
If there's certain music that you would really like her to have, you can create and share an Apple Music playlist. Another option is to set up a profile that shares your shared playlist and the songs you listen to. Once it's created, she can follow you.
Create and share Apple Music playlists
See what friends are listening to in Apple Music
Hoping that helps!