Welcome to iMore! It sounds to me like your cousin is paying for an individual Apple Music subscription, which is $9.99/month. Family subscriptions are $14.99/month. Have you tried to go through the subscription process yourself? I know a lot of times when my wife or kids, who are all on my Family Sharing plan, go to purchase an app that I've already bought they aren't notified that it's already been purchased until they actually tap on Purchase in the App Store.
If I'm wrong let me know and can look deeper into the problem.