They want to sell the Fire TV and Fire Stick because it will get you 1) used to talking to their remote, 2) hopefully get you searching and browsing around for all kinds of stuff - their selection is a fascinating mix of really new stuff and older stuff and b-movies, and it's often surprising what's covered by the prime subscription and what you have to pay extra for.

By getting you to spend more time in their interface, they can add things. E.g. Prime Music had me spending a lot of time checking out playlists, some of which may very well lead to purchases of albums where only a couple of songs are available for free.

It also gets them penetration for whenever they add shopping to these devices, and e.g. lets you talk to Echo and ask it to bring up the top five best new phones on your TV for you to choose fro.

Getting Prime on Chromecast would help their penetration of Prime, but it might hinder their overall service penetration by giving some people a reason not to get a Fire Stick. Especially when the Fire Stick/Fire TV and Chromecast are all cheap enough that there's little stopping people from getting both...

It may very well backfire for them, but there's all kinds of reasons for them to want to push the Fire TV devices hard.

But the Chromecast has no interface of it's own! If I use Netflix on Chromecast, I'm in the Netflix app on my tablet or phone. Amazon can still own it's interface.

I don't have enough ports on my TV for a Fire Stick and a Chromecast (and everything else) so I'm going to choose the one with the most options (Chromecast).

Yes, but if you're used to picking an app and casting from it, rather than picking a device and chosing apps on it, Amazon faces a much larger hurdle. On the Fire TV/Stick they control the UI. On Chromecast, they have to not only convince you to use Chromecast, but convince you to stay in their app.

They can lose a lot of users and still find it to be worth it for the extra engagement of the ones they did get.