Additionally, movie rentals are already covered by a prior agreement. The article you mention even says that Canal+ and other video services were already using this which is an option where you can allow purchases tied to an existing subscription payment method if you also allow for those purchases in the iTunes Store and integration into the Apple TV app.
There was no special deal for Amazon and trying to present it that way, is, imo, disingenuous. How can it be a special deal for Amazon if it existed prior to Amazon's app changes and other platforms were already making use of it?
Again, I have no idea what is possible on the Apple version, but I have subscriptions on other apps that go through the play store payment system. I'm not an expert on app store policies for payments, but I don't use any other apps that have subscription payment that don't go through the play store. I have a LastPass subscription which I purchased through their website, specifically because you have to go through the play store if you get it through the app.
On iOS though, apps need to use Apple's payment system for any digital good. Unless you're an Amazon Prime member renting a video.