This is an interesting development and it's hard to know what's going on behind the scenes here. Given that Apple is creating a tvOS App Store, it's hard to see how Amazon wouldn't be allowed to at least offer a non-default-installed Amazon Instant Video app that functionality equivalent to its iOS app (which allows for viewing of Prime content and non-Prime content already purchased outside the app but doesn't allow for in-app purchasing or renting of content). And I'm sure Apple would not extend the killer feature of the new Apple TV, Siri-powered "universal" search, to Amazon. That feature works only with a few popular Apple-selected apps: iTunes (of course), Netflix, Showtime, Hulu, and HBO. Clearly Amazon Prime, given its popularity, belongs in that club but I don't see Apple allowing it in there. So maybe Amazon decided being a 2nd-class citizen on Apple TV wasn't good enough for them.
On the other hand, perhaps Amazon is willfully withholding their app from Apple TV because it's a direct competitor to the Fire TV. That would seem odd, since Amazon Prime Instant Video really just serves as a selling point to its larger Prime initiative, and they should therefore want to make it as widely available as possible. But then, Amazon drug their feet on bringing their video app to Android and it's still not available for Android TV, except for Sony TVs that come with Android TV built-in. (A hacker was able to take the installer for Amazon Instant Video from such a Sony TV and distribute it so that other Android TV clients could access Amazon. It worked for awhile but then Amazon figured out what was happening and stopped streaming to those clients.) Furthermore, Amazon doesn't support the very popular Chromecast. Again, as with their standoff with Apple, it's hard to tell who's behind the Amazon - Google discord.