Android TV may look different from vanilla Android, but underneath that tiled interface is the same operating system you know and love. That means it's possible to sideload whatever APKs you want. The thing is, without a touchscreen, most of them are pretty useless.
Take the regular Amazon Instant Video app. Besides requiring that you have the general Amazon app installed first, it fails to provide a passable experience on Android TV (which lacks an officially sanctioned Instant Video app because Amazon). You can start and stop content just fine, but fast forwarding or rewinding is a non-starter.
You're better off using the Amazon Instant Video app intended for Sony TVs running Android TV. You can sideload it just like any other APK, and its customized interface unsurprisingly works well with remote controls. It's the same interface you can find on Samsung and LG's native smart TV apps.
To try things out for yourself, grab the APK from the link below. You can transfer it to an Android TV device (such as the Nexus Player) using, for example, ES File Explorer's send via network function.
The APK is signed by Amazon and upgrades your existing app. The cryptographic signature guarantees that the file is safe to install and was not tampered with in any way.
File name: com.amazon.aiv.us-1.35-4-minAPI20.apk
Version: 1.35 (4)