Its not clear to me that its a legal requirement to use a drivers license for this purpose, but its standard practice at a minimum. It IS a legal requirement for the receiving FFL to record where they received the gun from, and thats why they do this. Its a small due-diligence step to verify that its actually coming from the person you say it is.
This article may help with specifics. Probably helpful for the OP too:
Have you ever tried to send or receive a firearm but weren’t sure how to do it? Maybe you just bought a firearm online, or you need to ship a gun, and you’re wondering how to transfer it. You might have looked into transferring the gun and you learned that transfers the gun through an […]
rocketffl.com