Amazon Music is only available through Chromecast. There is an APK for Nvida shield tv you could sideload, but it's utter shit.

You can install the Amazon music app apk that you use on your phone. But ui won't be that great, but you can move around using your remote.

Main problem with this app is, you can use this app authorised to be used only on one device. You can either use that account in your phone or in tv, not in both.