Maybe there's a difference between Google TV and an Android TV, but my partner's dad can't buy/rent stuff on Prime Video through his tv, he has to use another device and then can watch it on his tv. He finds it very irksome, to say the least.
My understanding is that this came around by Google insisting on the 30% cut of purchases made in the Prime Video app on the tv, and Amazon told them to pound sand. I could of course be wrong, though.