Indeed. For some reason Amazon, appears to have a major axe to grind with Google only - I'd hazard it's their market dominance and the threat of the Play Store platform. Just makes it even more irritating. You can buy a Roku on Amazon too...Not a chromecast.[0]
I've always just cast the fullscreen video in Chrome to the Chromecast on the TV we have without a built-in Amazon "app". A bit clunky but works fine, unlike any of their mobile apps and non-Fire receivers.
[0]http://lifehacker.com/amazon-banning-the-chromecast-is-unbel...