According to its spec, Amazon Fire TV Stick 2nd Gen supports Bluetooth Low Energy (BLE). When however calling the Bluetooth API on Android it won't discover/connect to Bluetooth devices. Do these sticks have BLE built-in but are they not exposing it to the Android BLE API?
Amazon Fire TV 2nd Gen (the box, not the stick), does expose its BLE interface to Android apps though.
Bluetooth Low Energy (LE) was first supported in Bluetooth Version 4, but the specification says that device support for LE is optional. The page linked to in @MorrisonChang's comment above states that the Fire TV Stick Gen 2 has BT 4.1, but if you follow the "Supported Profiles" link to:
https://launchstudio.bluetooth.com/ListingDetails/26319
you will see that it does not support LE (which would be listed as "Low Energy Link Layer" under the "Supported Layers" section if it did). Interestingly, the Fire TV Gen 2 (the square box on a cable, not the stick), which was released the previous year, also has BT 4.1, but unlike the stick, it does support LE:
https://launchstudio.bluetooth.com/ListingDetails/17054
The first "Stick" type device to support LE is the Fire TV Stick 4K.