California does have a law that requires serial numbers on firearms built from 80% lowers. Federally, it does not ever need a serial number unless you are in the business of manufacturing firearms as a business, then you would also need a license. I have provided in another thread information on ATF rules on personal builds and serial numbers, etc., here.
Not part of this discussion, but comes up frequently when discussing our RKBA. Felons are “allowed” under federal law to possess antique black powder weapons or their replicas as they do not meet the definition of a firearm. They must be cartridge-less and muzzle-loading - further ATF detail in a pdf here.