I just paired my airpods with my Echo Dot, just so I could answer your question. Worked just fine.
- On the iPhone, open the Alexa app
- Tap Settings
- Tap the name of your dot (I have 2, so there's more than one name)
- Tap Bluetooth
- Tap Pair a New Device
- Hold the airpod case (open) near the Dot and hold the button for 5 seconds
- The Airpods will appear in the Alexa App's Bluetooth Devices screen
- Tap Airpods

