Under federal law, I think it's illegal for dealers to sell AP ammo that is made for handguns or fits in handguns, and of course there are "handguns" on the market that chamber both 5.56mm and 7.62 x 39mm.
I didn't think it's illegal for private citizens to keep any armor-piercing ammo (handgun OR rifle ammo) that they already owned, or acquired from a private party sale since then. At least not under Georgia state law or Federal law.
AP handgun ammo doesn't usually have a penetrating core. Instead the entire bullet is either made of an unusually hard material, although sometimes it's just copper. Even pure copper is harder than lead. Sometimes there's a steel jacket over a lead core. Often the bullet shape is key to the round's penetration abilities. Sharply-pointed bullets focus their energy on a small area, and a bullet's pointy tip can spread apart the fibers of soft body armor so it doesn't have to cut and break all of them to get through. Some handgun A.P. bullets have special outside lubrication, too.
There is a federal law that specifically defines what is "armor piercing" handgun ammo. It has to do with what the bullets are made of, not how they perform against soft body armor or metal plates.
The anti-gunners want to replace that definition with a performance standard, to ban any ammo of any caliber that will shove a bullet through a common Kevlar vest, class II or IIA. Naturally that will end up banning a lot of magnum pistol rounds and virtually all centerfire rifle rounds, even if the bullets are traditional soft-points with a lead core and a copper jacket.