No. As Jim said neither the SN or Manf is sent to the ATF/FBI. When you do your 4473. Just "Long Gun", "Handgun", or "Other".
UNLESS you are multiple purchaser then the FFL has to do a 3310.4 (Multiple Sales Report). ALL 3310.4s are in the FTS - DIRECTLY LINKING YOU WITH THE GUNs. The 3310.4 requires the FFL to report Type, SN, Manf, Model, Importer, Caliber, and Transfer Date -- along with all your pertinent info.
The FTS is publicly acknowledged. The NTC has a nice building on the Martinsburg, WV ranch/campus