No. California does not require scooter registration in the same way that it does for cars and other vehicles.
Note, however, that this is one of the few ways in which the law for vehicles like cars and the law for scooters deviates. Scooters are subject to all of the same traffic laws as cars. Scooter operators must obey all traffic laws and follow the rules of the road, as well as operate their scooters safely and prudently.