They eventually end up with the ATF.
However the fact that private party transfers still exist in much of the country means a several year old record of a firearm purchase is not reliable.
Making it a poor registry.
It is also specifically outlawed under FOPA for them to create a database or registry.
This is why part of the anti-gun strategy is to require registration, or all transfers go through an FFL resulting in such.
Often combined with making it a crime to not report lost or stolen firearms within a few days.
This means all guns on file should be at specific locations, and if they were to go ask for them, it would be a crime for them to say they lost a firearm beyond the days allowed for mandatory reporting, and they are not going to believe someone just happened to lose a gun within a couple days of asking for it.
It would also be a crime for an owner to file a false police report reporting a gun lost or stolen that was not lost or stolen, and so if found to still own that gun guilty of that offense.
The result is intended to be a society where the location of all guns is known, and excuses from owners for not being able to account for them are a crime.
The purpose of which is to implement additional controls over those firearms. Whether confiscation gradually of specific types or models or actions, or new hoops to jump through to remain a lawful owner or default to unlawful. Or confiscation on an individual basis using a variety of tactics, like ever increasing numbers of prohibiting offenses that turn someone into a prohibited person (CA has legislation right now adding even more misdemeanors and minor offenses, and already has included many beyond federal law like simple assault.)