Amazon's streaming is on Roku already, with support for Prime and rentals/purchases. I don't see why they'd have any problem with Chromecast. The Fire TV doesn't seem like a product where they intend to make money off of the hardware. It's just a means to get their content in front of more eyeballs.
Their big battle (in my opinion) is that Netflix has a whole lot of name recognition, and that's where making an Amazon streaming box can help. Right now people associate Amazon more with buying physical goods, and less with digital media where their main presence is with the Kindle brand.
I saw a Vizio smart TV that has a red "Netflix" button on the remote. That may change if Amazon becomes a legit competitor to Netflix, but Netflix definitely has a stronghold.
Their big battle (in my opinion) is that Netflix has a whole lot of name recognition, and that's where making an Amazon streaming box can help. Right now people associate Amazon more with buying physical goods, and less with digital media where their main presence is with the Kindle brand.