Just to add to this, the specific federal law which prohibits the keeping of a database of guns and their owners is the Firearm Owners Protection Act (Federal Law 18 U.S.C. 926). This is the relevant section from that law:
No such rule or regulation prescribed [by the Attorney General] after the date of the enactment of the Firearms Owners Protection Act may require that records required to be maintained under this chapter or any portion of the contents of such records, be recorded at or transferred to a facility owned, managed, or controlled by the United States or any State or any political subdivision thereof, nor that any system of registration of firearms, firearms owners, or firearms transactions or disposition be established. Nothing in this section expands or restricts the Secretary's authority to inquire into the disposition of any firearm in the course of a criminal investigation.






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