IF,,,the BATFE numbers the gun it will most likely have an 'ATF' prefix#. They will assign the number.
Or they may confiscate the gun.
I've had two such (handguns) come through that had ATF ser#'s on them. One was a stolen/recovered that was re#'d before being returned to the original owner.
Original ser# had been scrubbed by the thieves.
Even though the original was known (revolver was in NCIC), it was not restamped on the frame. The ATF# was used instead.
The fed law is simple,,Alteration, removal or obliteration of a firearms manufacturer placed serial number is unlawful.
Possession of a firearm with such is also unlawful.
Both are felonys under GCA'68.
Maybe the Colt P/Positive was never numbered in the first place. I'm sure some lunch box specials & parts walked out of the plant over the years.