These signifyin' monkey songs are almost a whole sub-genre in blues and early rock'n'roll. Chuck Berry did one of them, Jo Jo Gunne, which is basically the same story as this one and The Coasters did one called Run Red Run. It's a story of a monkey who grabs his human master's gun and it's often taken as a comment on how the black men would treat their "masters" when they got guns.
There are lots of signifyin' monkey songs in blues, but I think one of the best is The Monkey and the Baboon — which is about an ongoing feud by a couple of gamblers who try to psyche each other out with their signifyin'. I heard it from Paul Geremia who, if memory serves, said he learned it from Dr. John.
Personally, I see only a vague kinship between most of the signifyin' monkey songs I've hard and Straighten Up And Fly Right.