You can play more than one character in Super Mario Run, but besides Mario you must unlock the other five characters in the game. In the original version of Super Mario Run you can unlock Princess Peach, Luigi, Yoshi, Toad and Toadette. Once a new playable character is unlocked, you can switch to that character before starting any level in the World Tour section. Just tap the World you want to play, then tap the character icon next to the "Start" button to enter the character select screen. From here you can choose all the unlocked characters in the game.
How do I change my character in Super Mario Run?