Amazon’s 2nd-generation Fire TV Game Controller appears to have an accelerometer inside. When the controller is sitting unused, moving the controller causes it to begin searching and establishing a connection with the Fire TV. Pressing a button is not necessary to “wake” the controller. You simply have to move it. This isn’t the first time Amazon has included motion sensing in their Fire TV peripherals. The original bluetooth based voice remote includes an accelerometer as well, but it was never known if it was being used for anything. It seems as though Amazon uses the accelerometer in the game controller, and likely in the remotes, to ensure the device is ready to accept inputs before pushing any buttons. This likely improves battery life by not having the remote connected to the Fire TV at all times.
It’s unlikely there are, or have ever been, plans to use the game controller for motion gaming. While the Fire TV Game Controller does appear to have an accelerometer inside, it’s unlikely that it has a gyroscope, which is needed to more precisely detect orientation. Either way, if you store your game controller, and possibly remote, on a surface that sees movement throughout the day, you may want to consider keeping it in a more stable location to prevent it from constantly establishing a connection with your Fire TV.