states law as usual vary but only a few states have serial number database of guns. Ymmv if they will allow it or not.
So tl;dr, you only need register a serial if you made something like a machine gun and hold a federal firearms manufacturing license, and you can only make one or two versions of each design without changing things, and the serial number only really matters when selling or trading it, which you both are and aren't allowed to do depending on what the courts feel your intentions were. Otherwise the mechanical design of the gun can be freely sold to law enforcement or the military. Also the rules change once you actually get a manufacturing license (you are signing away some rights in the process, similar to signing away your right to not have an ID while driving when you get your driver's license) but if you get your manufacturing license you should probably have an actual lawyer.
This is really poor advice. The BATF considers just having the parts without the requisite Class III license (really a tax stamp) to be a felony.
The Hughes Amendment to the Firearm Owners Protection Act of 1986 prohibits civilians from owning any machine gun manufactured after May 1986.
All the other 5 categories of Class III firearms (SBRs, SBSs, Silencers, AOWs and Destructive Devices) can still be made, even by an individual, if he/she first applies for and receives permission to do so.
To do so, they must file an ATF Form 1 (maker form) and pay a $200 make tax fee.
A civilian can still legally own any machinegun that was created PRIOR to May, 1986 as long as they get approval on an ATF form 4.
Regardless, you should definitely speak to a lawyer before attempting anything like this, there are tons of local laws that you can break even if you are following state and federal law. And once you break a local law it becomes an illegal weapon and negates any legal standing you had with the state or federally.
The only ways to legally produce a new machine gun in the US right now are to pay thousands of dollars a year for a federal firearms license and special occupation taxpayer license to manufacture, or to otherwise receive explicit permission from a US government agency to get the ATF to approve a Form 1 application to make and register an NFA firearm. See 18 USC 922 (o) regarding machineguns not registered prior to 1986. (Wickard v Filburn dispensed with the notion of claiming to avoid federal law by keeping activity within one state.)