The movie is a parody of musical biopics, and it doesn't depict any real-life events or people. So, any animal companions Dewey Cox might have had are purely part of the movie's comedic portrayal.
Did Dewey cox really own a monkey?
The character Dewey Cox, from the movie "Walk Hard: The Dewey Cox Story", is a fictional character. While it's funny to think about him having a pet monkey, there's no indication that he did in the movie.