Depends how well they lock it down, Kindles are based on Linux and several of the recent models didn't have any jailbreaks available until winterbreak came out. Not sure if Amazon have managed to patch against it but I took the opportunity to jailbreak mine as soon as I found out about winterbreak.
Also worth noting that on the android based fire TV sticks, you can install android apps easily, not sure Amazon will give you that freedom on the Linux versions