I kind of wish Amazon and Google would make a truce and allow their services on each other's devices.
At least right now, Chromecast doesn't support Amazon video or music. I have an Echo so I listen to Amazon Music on that. Great sound but not the most reliable connection. I use my Sony BluRay to watch Amazon Video on my tv but the user interface is chunky and the remote control is a minor pain in the tuchas.
Amazon doesn't stream Youtube and neither does my BluRay.