I don't think they have to quote an account # they just do the whole "can you verify this" bullcrap.
Did you ask or did they say their agency name? They can do searches and stuff like "geographical search" trying to locate every possible number/address that's ever been tied to you, that's how they probably know your personal info. Don't worry.
Best advice at this point and time, you paid it off, let it go, live your life. Congratulations! Ahaha :D