How do I access Apple Music that is shared to me through a family share?

If you are sharing an Apple Music subscription you don't get the other person's library, you get to use the Apple Music subscription service in the same way that you would have had you taken an individual subscription. In short you should have access to all of your unhidden purchases, any locally added music, and can now add content from the Apple Music catalog as you please. That content, and anything located only in the cloud (other than purchases) stops working once the subscription is terminated.

It is possible for someone else to share an Apple Music playlist with you, such that you can download or stream the same tracks, but they have to actively do that, and you have to actively add their shared playlist to your library. It isn't something that happens automatically.

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The Organizer also needs to have the Family Music subscription, which costs more than the Individual Music subscription.

You can share Playlists with or without Family Sharing, but using Family Sharing to share the Family Subscription gives you your own version of Music and the songs are not shared.

How to share an Apple Music playlist on your iPhone or iPad - Apple Support