The shared Apple Music subscription, if indeed the Family tier of that subscription, wouldn’t appear as a listed subscription for the other group members. Instead, I believe it would be listed in the Family Sharing section of Setting on your devices. And you should be able to use the Apple Music directly, on your own Apple Account.
Do verify that the Apple Account as added to the Family Sharing group matches the active Apple Account on your device(s), else you wouldn’t get access to the service. Also, the adding of additional group members has to be confirmed by each group member.
If the above doesn’t apply to your situation, then consider contacting the Apple Support staff for further assistance, as they can check account details that may matter for this. Contact Apple for support and service (phone numbers) or start your support request online (chat/email/callback).