Actually, you DO need to register the moped. If it was made within the last 30years you must register it, although you don’t need a license for it. Having EVERYTHING helps though in case you get pulled over for something stupid, an old friend of mine had his moped impounded for not having registration or a license.

California law -
Moped or motorizesd bicycle producing less than 2HP and limited to under 20mph can be ridden without a license, or register the vehicle. Must be 16yrs old and ware a helmet.

Scooters are considered a motorcycle and require registration (license plate), insurance, motorcycle license or permit.