Correction: In Super Mario Bros., the 2D successors, and Super Mario 3D Land/World, Mario does indeed become Super Mario when he picks up a Super Mushroom. "Super Mario" is the name of his larger form.
Somewhat interestingly, "Mario Bros." now refers to the last name of the pair of plumber brothers, making their full names "Mario Mario" and "Luigi Mario" respectively. This wasn't originally the intent of the creators, but it's been slowly retconned over the history of the franchise.