Yes, it does press and hold down the home button on the firestick remote. Go to settings. Then pair other Bluetooth devices with it. (FYI) Although the Amazon Firestick is capable of. I do not recommend it if you are pairing for movies . Allot of pausing and source too slow error messages will pop up..

How do you connect Bluetooth to Amazon Fire Stick?
1.Switch on your Bluetooth speaker or headphones.
(Also, make sure the device is visible publicly.)
2. Go to settings in Amazon Firestick.
3. Click on Controllers and Bluetooth devices.

4. Select Other Bluetooth devices.

5. Wait for your headphones or speaker to appear in the Discovered Devices list. Once it does, select it and wait for pairing to complete.

Does Firestick 4K Max have Bluetooth?
Bluetooth 5 and Low Energy. Bluetooth speakers, headphones, video game controllers, and more that are also Bluetooth compatible. The included Alexa Voice Remote can control the Fire TV Stick 4K Max and certain functions (like power and volume) on a wide range of IR-enabled devices, such as TVs, soundbars, and A/V receivers.
How many Bluetooth devices can connect to Firestick?
How Many Bluetooth Headphones Can I Connect to a Firestick? Even though you can connect up to seven devices to the Amazon Fire TV Stick, you can only play audio through one Bluetooth device at a time.
does amazon fire stick support bluetooth headphones
If you have an Amazon Fire TV media streamer (like the Fire TV Cube or Fire TV Stick 4K) or a TV that uses Amazon’s Fire TV platform, you already have Bluetooth built in. The Fire TV platform lets Bluetooth devices like game controllers, keyboards, and headphones connect directly.