Store bought guns can usually be traced starting with the manufacturer through to the store to the customer. No real need for 'registration'.
Only if they already have the gun usually.
They need the make and serial number to search from the manufacturer to the first point of retail sale. And in may states that allow private sales the record ends a the first point of retail sale.
Even if the gun went through another FFL again, there is no way to tie it back to the original purchaser.
I think you would be very nieve to believe that 'big brother' dosn't keep track of names that get a background check with every set of serial numbers next to it but by all means, dont let me burst the bubble your living in.
Pretty tough since it in many states the serial number is never reported as part of the NICS check, or the make, caliber, or any thing much about the gun beyond long gun or handgun.
The forms to BATF and all records mandated by the Feds.
4473s oly have to h\be held by theFFL for 20 years ad can then be destroyed.
If the FFL terminates ther license any 4473s less than 20 years old go to BATFE to sit in file cabinets.
The FFKs bound book goes to BATFE when they end their licernse, to live in filing cabinets in WV.
I guaranty you, records of new firearm purchases are being held by the Federal Government.
Only if they are machine guns.
BATFE is prohibited every year from using any of their budget allocation to even try an computerize the old FFL records they have.
They live in file cabinets, ad must be searched by hand.
There is no way to take a persons name and search all the records.
Some states require registration, and others have been caught keeping background check data, but it is pretty far from a federal undertaking.