I’m not all that familiar with Washington State gun laws on registration, but this looks like the same thing that happens in Pennsylvania. I don’t think this is an issue of federal law at all, and the original posting that Dave Hardy links to misunderstands FOPA, as Dave points out.
Much like Pennsylvania, this information is likely obtained from state required forms, which don’t fall under the purview of the federal government. In Pennsylvania’s case, the “Record of Sale” is used to compile a computerized registry. If Washington State has a state form for gun purchases, it’s coming from that. Otherwise, Washington State is a partial Point-of-Contact state for Brady Act purposes, so it’s Washington State, not the FBI, which process background checks for every handgun sold. It’s also likely that system is tied in with a state “registry” as well. In Pennsylvania, a registry is supposed to be illegal by statute, but the State Supreme Court ruled that computerizing every Record of Sale was not a registry, because it wasn’t a complete record of firearm ownership. Of course, in the past, that hasn’t stopped the police from treating it as such, and confiscating guns at traffic stops for not being “legally registered” to the owner.