A link that I posted in the comments has provided some amusement as we warble our way merrily through a list of Partner Songs mainly from the American folk genre. Looking to expand the reportoire, I tried to think of some similar sounding songs to mesh but it's hard to track them down.
Jazz Web Page a list of songs that have similar chord progressions and tonal shifts
Songs with the common musical Pattern Desh "diatonic elaboration of static harmony"
Columbia Law School Musical Plagiarism Project - Song List
allows you to hear the superimposition of melodies
eg My Sweet Lord/He's so Fine
Disturbingly Similar Songs
In the meantime I should study my counterpoint