The phrase "singin' Dow Wah Diddy Diddy Dum Diddy Dum" does not refer to anything being sung by any of the characters IN the song (the girl and the guy). It simply provides a singable, repetitive, strongly rhythmic phrase in much the same way as....

"Fa la la la laaaa, la la laaaa la"

or

"How I wish I was in Sherbrooke now...."


The purposes of such phrases in songs (whether or not they mean anything) is to provide a rhythmic section that an audience can sing along with. They can be absolutely meaningless...such as "Fa la la la la" or "Hi, Ho, the Derry-O"...or they can have meaning, such as the line from the Stan Rogers song.