If your FFL is sending your gun to another FFL, no license is needed.
If you're shipping a gun directly to an FFL without using another FFL to send it, then the receiving FFL needs a way to book the gun in. The easiest way is to ask for a driver's license as it has the name and address of record of you, the sender. Same reason you have to give your license to the guy behind the counter when you buy a gun...they need to identify you so they can book it in to the correct name and address, and book it back out when they ship it back to you.