The UX for this is very awkward (unless I'm missing something) - the "store" and the "player" are completely disjoint experiences with only a few marketing links between them. You can't actually add new (free) "Prime Music" to your collection from the player - you have to jump to the "store" to discover the music you want and add it to your cloud library. Nor can you play full songs from the "store", even if you have Prime and they are free - you are limited to 30 second samples.
The amazing thing about Rdio and Spotify, and what makes streaming services different than their MP3 purchasing predecessors, is that you can think "I want to listen to Kanye", type "Kanye" in to the search bar, press enter and listen. Amazon is making this whole process very awkward by adding a "buy" step (even if no money is changing hands).
The reason that Spotify is disruptive isn't that the music is paid for via subscription pricing - it's that it eliminates the distinction between "music I own" and "all of the music ever recorded".