The Max app on the Apple TV box should give you access to their service when signing in with a valid account.
Besides an in-app subscribe option for new subscribers, the app should also have a Sign In option for existing accounts. If the app has already been signed in to, then you can sign out, and sign in with the AT&T account.
Sign in to Max - Max Help Center
↳ Which device are you using?﹥TV device﹥Scan QR, or use your remote
If you are not using the Apple TV box, then the Max app on your smart TV should similarly have a Sign In option for existing accounts. This would not have anything to do with subscriptions through Apple, nor the Apple TV app, nor your Apple ID account.