What did Amazon do?
Google Cast players can't stream Amazon content because Amazon hasn't implemented the Cast SDK in their app.
Amazon doesn't implement the Cast SDK because their flavor of Android eschews Google Play Services which is required for the Cast SDK on Android.
Google doesn't license the Google Play Services to Amazon because....
We can go around for a while...
Because Google refused to allow Amazon to cast to Chromecasts.
There was once a time when the Amazon Video and Netflix apps on Amazon Kindle Fire could stream to Chromecast, Google then changed their protocol to prevent that (you now need to use their proprietary library for it).
Amazon reacted by removing the Chromecast from their store.
That’s a funny claim, considering Google intentionally has changed the protocol several times, to prevent any third party implementation of the protocol – including Amazons, and many Open Source libraries for it.
I was building an open source library for it when these changes happened, and I can tell you, it’s not pretty.
The only way to actually reliably stream to Chromecast, and to receive Chromecast streams, is with Google’s proprietary library for it. On Android, this is included in the Google Play Services.
Obviously, on Amazon Kindles, these aren’t preinstalled (due to their restrictive terms), so Amazon had their own implementation (which, as mentioned, Google acted against)