Hey there, Whathaveudone,
Thanks so much for using Apple Support Communities.
We are happy to help you get this sorted out in the best way possible. It sounds like your sister's Apple ID may be signed in to your mom's iPhone for everything, rather than just Apple TV. This would include App Store, iTunes Store and could also include Messages or FaceTime. The best way to accomplish what they're trying to do would be to utilize Family Sharing which is a supported, seamless way to share things like Apple TV with your friends and family.
Here is how to set up Family Sharing, your sister would want to set this up on her device: Set up Family Sharing - Apple Support. Afterward, your sister can add your mom to her Family Sharing group and share the Apple TV+ subscription.
We hope this does the trick for you!
Cheers.