The music on the existing Amazon Prime Music can be downloaded to one (I think actually up to a couple) of registered devices, definitely including Amazon Fires (of course) as well as iPhones and iPads. I'm sure a bunch of other devices as well. It's not nearly as ginormous a library as Spotify, but especially if you're focusing on more vintage content it's pretty good. You can't rip this content beyond the authorized devices.
Amazon is in the process of launching a new music product designed to compete with Spotify and Apple with a very (but not completely) comprehensive library. That product will apparently require an incremental subscription fee on top of regular prime. (The current Amazon Prime Music is included with a regular Amazon Prime subscription). It will provide all the same download/rip rights as Apple and Spotify.