I think this is generally how it works now, you can only have 1 "Real ID" from 1 state, and that state becomes your "domicile" as @45Doll says. So to the OP, what DL do you have? That determines what laws you must follow to purchase a firearm, even when purchased out of state. If you have NJ DL and wish to purchase in FL, then the FL dealer must follow NJ law in order to sell to you. IANAL but that's my understanding.
So when you buy a firearm in FL and fill out the 4473 Florida law demands that you use a FL driver's license for identification?
Also, does FL make you fill out any additional forms like our P2P for handguns?