In most of the US, and I suspect for most of the population, there is no such thing as registering guns (that would be akin to registering laser printers, especially those high capacity ones).
Interesting choice of example, as it happens there is in fact a "printer registry", where documents can be traced back to the device that produced them: https://w2.eff.org/Privacy/printers/docucolor/
(it's just that the average consumer doesn't know about it)
To the extent there's any tracking of printers, it would at most tell you which store it was sold from, there isn't any required serial number recording done at checkout (well, as of when I last bought a color printer in 2006).
What you're referring to would be akin to comparing a bullet found at the scene of a crime with one fired from a seized weapon, albeit easier; a comparison system. Much less of a privacy concern, without any required "registration" involved.
In most of the US, and I suspect for most of the population, there is no such thing as registering guns (that would be akin to registering laser printers, especially those high capacity ones).
On the other hand it's illegal for a drug user to possess a firearm, 18 USC ยง 922(g): http://www.law.cornell.edu/uscode/text/18/922