Hi, and welcome to the Apple (user-to-user) Community!
Re: "I added my mom to my Apple Music family plan and she can't play any music -what did I miss?"
Check if you added your mom here too: Add a family member to your shared subscriptions - Apple Support
If you wish: Purchase sharing is a feature that you can use after you set up Family Sharing.
(NB: Seems one adult in the family — the family organizer — pays for everyone's purchases.)
When you turn on purchase sharing, everyone in your family gets access to the apps, music, films, TV shows and books that family members buy from Apple: How to share apps and purchases with your family - Apple Support
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Re: "when she attempts to play any music she received an error message saying she can't access the content".
For your mom:
Open the Apple Music app and tap Listen Now.
In the top-right corner, tap the photo icon.
Sign in with your Apple ID and password.
Everyone in Family Sharing automatically has their own Music Library.
They can add whatever music they wish to listen to.
Previous purchases may be available in the "Library" in the Music app
or how to: Redownload music - Apple Support
If you can't find your family's shared apps and content - Apple Support
How to download apps and content that your family members purchased - Apple Support
If needed:
Music Support Website: (info / support / contact options etc.)
Music - Official Apple Support
All the best :-)