Hi,
You can't currently share complete Apple Music libraries, but, since you have an Apple Music Family subscription, you can add your mom to a Family Sharing group and share playlists.
To set up a Family Sharing group, follow this guide: Set up Family Sharing - Apple Support. Once your mom accepts your Family Sharing invitation, you'll both be able to access Apple Music with ease.
You can also share playlists with songs you both like. For more information on how to do that, watch this video: https://www.youtube.com/watch?v=c6wRCTPD8M4
For more details: Share Apple and App Store subscriptions with family members on iPhone - Apple Support.