FAA: Helicopter means a rotorcraft that, for its horizontal motion, depends principally on its engine driven rotors.
For a coaxial helicopter the category is Rotorcraft and the class is Helicopter.
Your certificate would read; Private Pilot or Commercial Pilot, Rotorcraft-Helicopter
Any Rotorcraft- Helicopter CFI could legally give flight instruction in a coaxial helicopter. That does not mean they would.
Having an unpowered rotor is part of the light sport aircraft requirements.
There are many experimental, amateur built helicopters.