I've always added my Soc Security # to the ATF/background check forms just to be sure I'm not confused with another person or there is a big mix-up.
I have no criminal records or problems and most gun shop/pawn store trips are easy-peezy, lemon-squeezy,
ps; the only minor gripe I've ever had is gun shop staff who ask for all names, addresses etc be spelled out. No abbreviations. That may jam up a background check.