Some states do require some form of firearm registration, but it is on the state level. There is no federal firearm registration.
Arizona does not require firearm registration.
The paperwork you fill out when you purchase a gun, The ATF Form 4473, or Firearms Transaction Record, is simply a method of ensuring that Federal Firearms Licensees (aka Gun Store Owners) are doing their due diligence not to sell guns to people who shouldn’t have them. The accompanying FBI background check is part of that process.
The paperwork is required to be stored on-site at the gun store, but it is not recorded in a federal database. If something were to happen to your firearm, the authorities can contact the gun store and ask to see what’s on the paperwork. However, since gifting a firearm to another person is legal in most states (with some exceptions) the 4473 paperwork can only tell so much.