It's Google not Amazon.
https://developers.google.com/cast/
Amazon is the one banning it.
Amazon is the one stopping casting from their devices. Not Google. Google Cast is free to setup for. https://developers.google.com/cast/docs/design_checklist/
Nothing needs to be sold through Google Cast, no royalties, etc. All Amazon has to do is SUPPORT it. Or atleast allow homegrown solutions to thrive.
I agree this is a stupid 'feud' that only hurts consumers and I have my likes and dislikes for both companies but if you cannot see the difference here then I'm not sure how to help.