No. It is not.
It is a database of who is eligible or ineligible to own firearms.
It is not a data base of what firearms are owned by whom.
They NICS system is prohibited from retaining information on what firearms were purchased by what individuals.
To trace firearms, the ATF traces a firearm by serial number from the manufacturer to the dealer who sold it. The dealers are required to keep records of to whom they sold firearms, and respond to traces on individual firearms.
State or local agencies may have their own gun registration system, but NICS is not a gun registration system.
They retain the transaction info for some 6 months (they say) as a quality control measure.
The NRA has brough suit against NICS for building a database of gunowners using the "audit log." NRA vs. Reno