Amazon has monopolistic power over this because set-top boxes aren’t a primary component of Amazon’s sales; it can turn away buyers of Apple TV and Chromecast without materially affecting its overall business sales numbers.
It’s perfectly logical for Amazon to only feature TV products that work with its services. This policy is clear-cut for the current-gen Apple TV — no Prime support, no sales. However, this reasoning does not really apply to the Chromecast. The Chromecast SDK is offered to anyone to implement and arguably is ‘easily compatible’ with Amazon Prime Video if Amazon developed the integration. This nuance doesn’t seem to have been addressed in the announcement email.
It’s also not clear whether this policy applies to the new Apple TV, where Amazon could make an app and get its content on the box. On this front, I suppose we’ll find out in October when the product launches.