When the writers* behind 1993's Super Mario Brothers: The Movie were establishing the rules of the film's universe (after all, the movie does take some creative liberties with the Super Mario Brothers canon) they decided that the heroes, Mario and Luigi, needed last names. At some point it was decided that since they are known as the Mario Brothers, then their last name must be Mario (as in, if the heroes were Bob and Ted Jones, they'd be known as the Jones Brothers). Hence, the characters were named Mario Mario and Luigi Mario. Adding credibility to these names was the fact that Nintendo themselves signed off on the idea, proclaiming to all in Nintendo Power magazine that everyone's favorite plumbers had a last name, and it just happened to be the same as the lead brother's first name.

Then once the movie was released it bombed horribly and Nintendo sought to erase all memory of the doomed film from the public consciousness. Mario and Luigi's last names were stripped away, relegated to the memory hole, and the company has never made use of them again.

* Yes, contrary to popular belief the movie did have a writing staff.


References:
Nintendo Power circa summer 1993
http://www.classicgaming.com/tmk/