I once and not so long ago worked on the other side of the local gun store counter. Doing such as a hired clerk I had to be finger printed and IDed.
So then I sold guns to guys like you, and made calls to NICS.
NICS gets the info you put on the 4473 form EXCEPT the gun is offered as these 3 and only these 3 selections.
Long gun meaning a rifle or a shot gun.
Hand gun meaning a revolver.
Pistol meaning a semi auto.
No name, year, or model, and no serial numbers go to NICS.
So if NICS keeps any records on the sly, all they can know is you bought one or more of the above guns.
I bet you can't guess which of these 3 types I have, and so neither can NICS.
My best guess as to the 'heard the serial number read back over the phone was military speak read back on the date, and other related info given because talking to NICS is like talking in the military, and to a robot at that!
You heard a garbled message, which may have sounded like a serial number, or things have changed recently and I would be wrong.
Another numeral alfabit is the store code which ID's the call is real.
I am 99% sure you never really heard the serial number, but I can see how you might think so. But in that event you should have heard what specific gun model, and the makers Name.
A serial number with out the maker and the model is useless.