In the US version of Mario 2, all you have to do is make it to the end with any character to win. In the Japanese Doki Doki Panic on which it is based, you actually need to get to the end with all 4 characters before you’ve beaten the game.
The different characters do have strengths and weaknesses: differences in run speed, item pickup speed, and jump mechanics (and Toad actually runs faster when he’s holding an item) but you are correct in that for purposes simply amounting to no more than getting through each level to the end, using Peach/Toadstool is the easiest.