Not on a passport card. It's just like any other id...plastic with holograms, so you're not scratching anything in with pen or pencil. Plus, there is no address anywhere, just Surname, Given Name, Sex, DOB, POB, Passport #, Issue Date, Expire Date, and a photo. So that and a local state registration should, by all rights, be legal anywhere in the U.S. The Reg alone has no photo or DOB, so that violates the Brady Act.
Now, whether or not this stuff just gets put on the 4473 or goes to the NICS I don't really know. I do know after looking up a DUI I had in Miami many years ago, that there is a guy with the same name, different race, different age, with multiple felonies, so could be mistaken identity. It's never been an issue before, but CA is a also a POC, so the NICS doesn't make the final call. I wouldn't suspect that to be the issue, but I trust the government about as far as I can throw them, so there's that.