"I don't have the full data set, if any is the right term. Generally speaking, it's a Bluetooth controller," Mike Frazzini, VP of Amazon Games told Engadget today when asked about the Fire TV's controller support. "And the same thing vice-versa. Third-party controllers can work on Fire TV as well."
According to a mandatory configuration file for all Amazon Fire TV game developers listed on Amazon's site, developers must note if the app "supports a game controller, either the Amazon Fire game controller or a Bluetooth game controller from another manufacturer." While Amazon doesn't confirm official Fire TV support with the game pads, other Bluetooth controllers like Nintendo's Wii Remote and Sony's DualShock 4 could feasibly work with some games if developers so choose.
Likewise, while the Amazon Fire TV supports up to seven Bluetooth game controllers simultaneously, software development kit documentation notes that "only four of those controllers are assigned to player numbers." This seemingly caps local multiplayer gaming on Amazon Fire TV at four players at a given moment, which may not be an issue since gaming is more or less a "bonus" for the streaming device.