I'm a Prime subscriber but can't watch Prime videos (TV and movies) on my regular Android device, although if I had an Amazon Kindle device I could watch them there. (And Kindle devices are an Amazon fork of Android, thus benefiting from Google's work.)
When it comes to physical streaming devices, I can use my Prime membership to buy an Amazon Fire TV, but I can't buy an Apple TV or a Google Chromecast. I could understand if Amazon wants to sell only their own device, but they do sell third-party Roku devices ("ships and sold by Amazon.com"), which means they are selectively targeting certain products.
This is pretty clearly a case of Amazon trying to use its leverage to improve its position in the streaming hardware and streaming content markets.