Amazon Prime claims that you can "own" the movie. Problem is Prime is still just a streaming service. It's false advertising and the reason I don't use Prime for movies. If I "buy" a movie, I expect to be able to d/l to a portable drive so I can watch it when I don't have a data connection. If I subscribe to streaming service, I won't have that expecation.
I think you mean Amazon Video, the division that sells content for download and purchase, not Amazon Prime which actually is a streaming service similar to Netflix. However, by this definition, you are buying movies from Amazon Video; not just streaming. Any video content that you purchase from Amazon can be downloaded to your Android or iOS device (including an external microSD Card in the case of the former) with the Amazon Video app for later playback offline; no data connection required. We do this regularly to watch movies from Amazon while on a flight, in a car with no wifi, etc. You can even download Amazon Prime video (which you do not own) and play it offline for a certain period of time, which I believe is 30-45 days from the time of download; quite reasonable for content that you do not own, IMO.
What you cannot do is play it back on any device with a player of your choice. Amazon Video, just like Apple's iTunes, Google Play, Vudu, UltraViolet partners, etc., places DRM on all content that they sell, and it will only play on authorized devices and software.
- Stealth Dave