There may be a little confusion here.
There are several jurisdictions involved in firearms laws.
The top level is the Federal Government. The Gun Control Act of 1968 (GCA 68) established the system of firearms dealers as we know them today, and restricted interstate transfers of firearms to Federally licensed dealers (and yes, collectors, but let's not get into the C&R FFL right now). Changes in the law in 1986 under the Gun Owner's Protection Act (GOPA) permit you to purchase long guns from a dealer in another state, but not handguns.
The Federal paperwork is the Form 4473. It does not constitute "registration," as there is no way to find out what firearms you have by means of the 4473, as those forms are, by law, not to be compiled by the Feds in any way. The only use they have is to allow forward tracing, i.e., manufacturer/importer to dealer to customer. There the paper trail stops, at least formally, for the Feds. (Yes, I know about the multiple handgun purchase form, but let's set that aside too for now).
The next level of jurisdiction is the State level. Here's where the ugly head of "registration" raises itself from the slime. Many states, e.g., Michigan, New York, require that you complete a form (at least for handguns) that is entered into a statewide database whenever you purchase a firearm. That allows the state to know exactly what you have, so long as it came via a dealer. In some states with this system you must also report private transfers. This is registration.
Finally, in some areas we have local jurisdictional intrusion into our rights. For example, Cleveland Heights, Ohio required that all handguns be registered with the City (this, in the absence of any State law or requirement). I once inquired of the Sgt in charge of registration as to the nature of the records they kept; he showed me a shoebox with a handful of 3x5 cards. The PD recognized that the law was ignored, and they had no interest in enforcing it anyway (or, so I inferred). Ohio recently passed a preemption law that requires local jurisdictions to back off to the State level in terms of firearms regulation. It remains to be seen if Cleveland, for example, gives up its AW ban in conformance with the new state law.
As has been indicated earlier, "Your mileage will vary." It's your responsibility to know ahd understand the laws that are applicable to you. The NRA-ILA has a compendium that's a place to start. Most State Rifle & Pistol Associations will also have legal guidelines for their state's citizens.
Here in Ohio we are relatively vigilant, with "shall-issue" carry, the newly-enacted castle doctrine, no state registration, no state restrictions on firearms type, no restrictions on private sales, and so on.
Pity the poor Californian, by contrast.
Regards,
Walt