All of the “buy back” programs I ever read about is a no questions asked purchase of guns. So no there is no ID of the seller.
Several but not all of the buy backs I have read about said that there would be ballistics test of the guns to find if they were used in a crime.
This has always seemed a stupid waste of time to me because of the no questions asked policy. I also thought that is would be next to impossible because of the number of guns these programs could produce, the man hours required for a ballistics tech to check all of the guns would be enormous. Not to mention that the chances of a gun used in a crime showing up at one of these buy backs is incredibly small.
On some models, the marks left on bullets are so similar, that all a ballistic test can tell you is the caliber and generic model, such as “it was fired from a 9mm Glock”.
I recall one case where several officers were firing during an altercation. Ballistics could not determine which gun the fatal bullet came from.