And even on iOS, Amazon still has video apps, they just don't allow you to buy or rent directly within the app: https://itunes.apple.com/app/amazon-video/id545519333
There's nothing stopping Amazon from supporting Apple TV or Chromecast, except itself.
Chromecast doesn't have any special connection to Google Play. Anyone can cast whatever they want. Amazon Prime Instant Video can already be cast, but the user experience is so bad that you probably don't want to.
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Well, sure, you pay Google for the device -- that's fairly direct monetization.
And it increases the utility of (and thus the demand for) content from and subscriptions to Google's audio/video services (whose mobile apps all support Chromecast), so its monetized indirectly that way.
Other than that, its adoption drives app developer adoption and user familiarity for Google Cast, which, aside from its use in Chromecast, is a key feature of Android TV, which Google monetizes not just by selling hardware direct to consumers, but also via its cut of apps in the Play Store, and by (presumably) getting paid by Smart TV manufacturers to use Android TV as their Smart TV OS.
It probably leverages Android in some fashion also, though good iOS apps have chrome cast support as well.
It must be another reason... But what?
Apple's refusal to even allow links to external stores is petty and obnoxious, but Amazon is essentially playing the same game: if you won't let us play the way we want, we're going to make life difficult for our own customers while clutching our pearls and telling everyone that you made us do it.