My understanding is that you can cast to a Chromecast from a Kindle device if the app implements such a capability, but that you need either the Chromecast app (barred from Amazon devices) or the Chrome web browser (also barred from Amazon devices) in order to set up a Chromecast.
Ergo, you can't buy an Amazon device, buy a Chromecast, and get them to work together without another outside device. And there is no technical reason for this limitation.
Ergo, you can't buy an Amazon device, buy a Chromecast, and get them to work together without another outside device. And there is no technical reason for this limitation.