No, you need two copies of the game and you use the save data of whoever chooses to play as Mario.
However, I would love it if there was at least a way to spoof NSBM2's form of "local" multiplayer over the internet to add unofficial online multiplayer to many games that lack it! (Similar to the unreleased online multiplayer mod of NSMBW that chadders showed off a while back.)
I would also love the same thing for DS games (at least when playing in an emulator). I would absolutely die for NSMB Mario Vs. Luigi battles with hundreds of online opponents available to completely pwn play with! Beating my friends all the time in this mode was how I ended up with my nickname/username about 9 years ago! It would be amazing to have online multiplayer in the NSMB mode that made me become I pwned U!