Can you only play as Sonic in the one-player mode of Sonic the Hedgehog 2

Yes. Sonic is the only playable character in Sonic the Hedgehog 2 Single Player. The only other single player character you can play as is Knuckles. To play as Knuckles, you need Sonic and Knuckles and attach it to the game card of Sonic the Hedgehog 2.