To play "online," players typically use software that simulates a local environment by streaming the host's screen and sending the second player's controller inputs back. How To Play Cuphead Online Multiplayer | Parsec Blog
There is no official "online" menu or matchmaking within the game itself. Cuphead online
By default, you can play as Cuphead and Mugman (or Ms. Chalice with the DLC) on the same system. To play "online," players typically use software that