The iTunes application for Windows says Amazon Music is available for both iPhone and iPad. However when following the link from the iPhone 5S says "Amazon Music is only available for the iPad"
Ok, it's solved with a workaround. Go to iTunes on Windows -> Go to Amazon Music for iPhone -> get it. If you have auto-sync new apps on it will be automatically downloaded and installed on you iphone, even when the itunes for ios says it's not available.
In any case, my application got downloaded too in:
C:\Users\MyUser\Music\iTunes\iTunes Media\Mobile Applications\Amazon Music 5.5.3.ipa
Bad Apple, blocking its competitors.
How to enable automatic downloads:
More info: probably that's a glitch, other users also experienced the issue before:
Note that when I did search for "amazon" on iTunes on iPhone didn't find anything. Now it does, so probably a server bug or something. After you install something from Amazon the other apps show up as well.