This has existed for years, so I'm catching up here - though I do note that it has numerous updates over the last year, so it's actively developed still. From the Store description:
Unoffical Amazon Music streaming app, not affiliated with Amazon. You can now play any Prime music, in your library or not. Also supports Prime Stations.
Your login and security question answers are transmitted only to Amazon's server via a secure connection. Topperware, Microsoft or any other third party NEVER sees your personal details.
Songs are played in the background while continuing to use your Windows Phone and use the Music player controls to pause or skip tracks.
In trial mode you can play 10 tracks then you will be locked out for 30 minutes until you can play anymore tracks, once purchased playback is unlimited. Please purchase the app to fund future development.
Tracks can be downloaded from the album or artist details page, tracks which you have purchased or have uploaded can be downloaded in the background. Prime tracks can only be downloaded while the app is in the foreground, to please keep the app running while downloading.
Downloaded tracks can only be played by this app.
Some promo screenshots:
Playing Prime Music, and note the use of the standard Windows media system, here popped at the top of the display. This means full background playback; (right) browsing artists and albums from the Prime directory...
There's support for playlists too, if you want to go call in on Amazon Prime Music.... Note again that these are promo shots, I'd hate you to think that I had an X Factor list...!!
There's a free trial in the Store, limited to 10 tracks at a time, though it's only 79p in an IAP to unlock full functions, so it's not expensive. At least not compared to the £80 or so a year to be a Prime member in the first place?