The default player 1 controls (up, down, left, right), also control the menus of the second player if selected. The second player controls only affect the second player menu.

Video showing the issue:
https://www.youtube.com/watch?v=HJVdF7TUMuY

I believe the character selection menus should be tied to the individual controls of the players. On the other hand, this could be used to mess with people when playing local coop...