At first I thought it was weird that Mario's last name would be identical to his first, but I don't think it really matters. He's a video game character; some cartoon characters don't even have a surname. (And Luigi Mario has a nice ring to it) Also, 'Mario' is not necessarily the bros' official surname; it's just the most widely accepted.
I don't really understand why Toadsworth says "Master Mario," but I don't spend my time investigating a video game character's etiquette.