LORELAI: Hey, what do you know about this town loner guy?
LUKE: Same as everyone. Just kind of skulks around with that backpack, never smiles.
LORELAI: Does he also make cheeseburgers and secretly harbor a desire to wear a backwards baseball cap? … [sings] They’re cousins, identical cousins . . .
Lorelai sings the theme song to The Patty Duke Show, a sitcom which ran from 1963 to 1966, and featured teenage star Patty Duke, previously mentioned, as “identical cousins” Patty and Cathy Lane, who looked the same, but had opposing tastes and personalities – so not really a very good reference if she is trying to imply that Luke and the Town Loner are very similar in their demeanour and habits.
The Annotated Gilmore Girls 