Yep, I have it for Android from Google Play Store, and I distinctly recall the iPhone of course has it too from their store. I used to use it on an iPhone 4s.
Yeah I got the Prime for buying their FirePhone (since sold) and I love the 2 day ship, but notice that Prime is free as long as you keep it on their Cloud - you cannot download Prime music to your device. However, I think I did for some, I never replicated it since, but I did manage a few songs. I am a schmuck and pay for my music though so when I listen to Prime songs, I want to download them so I'll pay. Many I delete.
It is just basically called Amazon Music. It will play your stuff in the Library on the Phone itself or in your Library that you may have uploaded to the Amazon Cloud. Just in case this matters or helps, I upload the entire Library to the Amazon Cloud and pay them the 25 bucks a year for it. Observation here is that the device obviously works better overall than the Cloud. Even if your connection is spot on, using a Cloud just rarely could cause a hiccup since you are relying on the Cloud's far off server farm.
And I did check just now on my phone that the app does show Prime Music. So you should be able to hook up the phone to the 3.5mm jack or just use the Bluetooth and play. Just make sure you have a data plan when roaming between the WiFis. AT&T gives me what I want to settle that option, but personally I use the USB flash drives.
I saw Bluetooth cause the app to have issues during playback on the iPhone and when I had the FirePhone. the Android (HTC) has been perfect since.
More than you want to know, but in case anyone can benefit and while I am typing, heck why not.