What classification of aerial vehicle would a manned quadcopter or octocopter be?
From 14 CFR 1.1:

Helicopter means a rotorcraft that, for its horizontal motion, depends principally on its engine-driven rotors.
Rotorcraft means a heavier-than-air aircraft that depends principally for its support in flight on the lift generated by one or more rotors.
Aircraft means a device that is used or intended to be used for flight in the air.

A manned quad- or octocopter seems to fit the definition for “helicopter”, assuming you mean the kind that have no fixed wing.