Lyric discussion by TheJHam 

Cover art for My Monkey lyrics by Jonathan Coulton

It seems to me that the most obvious interpretation of this is that he is just referring to himself in the 3rd person. Either that or he actually has a monkey. Either/or.

i think that the monkey is his subconscious desires/impulses. the uncivilized/evolved part of his brain. he loves her but he still needs and want's thing himself