These two songs seem to be versions of a common story in which one party gives the other a series of gifts, each of which may turn out to be faulty in some way.

Mockingbird (e.g. recording by Carly Simon & James Taylor)

If that mocking bird don't sing
I'm going to give her a diamond ring...

(Childrens skipping rhyme)
Nebuchadnezzar, king of the Jews
Bought his wife a pair of shoes..

Is there a single 'name' for this group of songs? Are there many variants?

Are they related to the Riddle song or the Lovers tasks (Scarborough fair group of songs)?