There are several playable characters besides Mario in Super Mario Run, but you have to unlock them in the full version of the game to access them. You can also play as Luigi, Yoshi, Princess Peach and Toadette. Each character comes with their own special ability. Luigi can jump the highest using the long press jump, like Mario. Here's how to unlock Luigi in Super Mario Run:
To unlock Luigi you will need to compete in Toad Rally to earn Toads. Luigi requires 150 Green Toads and 150 Purple Toads to unlock his house in the building section. Luigi's house is located in the Special section of the Shop screen.