Then, of course, there are songs written with two melodies meant to be sung at the same time. The first one that came to mind was "I Hear Music But There's No One There" by Irving Berlin. Its official name is "You're Just in Love (I Wonder Why)," written for the 1950 musical Call Me Madam. Frank Loesser wrote lots of such songs - "Fuge for Tinhorns" from Guys and Dolls, for example (I got the horse right here...).
It's fun to sing such "partner songs - the confusion is part of the fun.
-Joe-