You can watch Netflix with ease on a Chromecast, but there's no easy way to stream Amazon Video since Amazon has its own streaming dongle.
There is a workaround, though, as long as you keep a computer with the Chrome browser handy while you're watching on your TV.

Open the browser and navigate to Amazon Video and whatever you'd like to watch. Select the menu on the top right (). Select "Cast" and a pop-up window will ask you to select the device to which you'd like to Cast.
The downside is that you'll have to go into the browser to control the video.

The quality is also not ideal. Google is testing a better experience, though.

Go to chrome://flags/#media-remoting, enable the highlighted flag, then restart the Chrome browser, type in the URL of a video you'd like to play, select Cast in the Chrome menu, and make sure to expand it to full screen.

https://www.pcmag.com/feature/326584/23-things-you-didn-t-know-your-chromecast-could-do/18

これいいよ