I've had Amazon video and music apps on my Android devices ever since I got prime, and they've worked without a problem. I mean, aside from not working with chromecast (but I don't do that often anyhow).

Maybe they've got a deal with Amazon to pre-load the app? I did sideload the Amazon app store.

Each Android TV OEM has to have a separate agreement to permit use of the app. So far only Sony and Nvidia have done so. The sideloaded version doesn't work on other devices.

Ah, so you're just talking about Android TV devices? I thought that the Shield was a kind of portable game system, and that it might obey the same "rules" as phones do.