There's a much tighter serial number system used for guns, all the way to a BATF Form 4473 (https://www.atf.gov/files/forms/download/atf-f-4473-1.pdf) that's required for the first retail sale.

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.