I doubt LA has any type of gun registration. Most of the southern states have no state control process for buying guns. You go through the regualr federal background check when you buy a gun. The only record of the transaction is paperwork at the dealer, which is, by law, not considered a registration, and can only be used by the government to investigate specific crimes.
In most southern states, if someone gives or leaves you a gun, you don't have to register it, and it is legal to keep as long as you are legal to own it, i.e., you are not a felon.
The media describing legally owned guns as "properly registered" is just part of a misperception that guns are (or should be) controlled everywhere.