I think what happened is they designed his appearance before deciding on a name, so they had to choose a name that fit his appearance.
His appearance was dictated by graphical limitations. Overalls because it made it easier to animate arms moving. A hat because hair is tricky on 8-bit systems. A moustache because a mouth is also difficult. All this adds up to a carpenter, which was changed to plumber when Mario Bros. came out in arcades (which took place in the sewers, rather than a construction site), and I guess the moustache in particular they thought an Italian look; therefore, Italian name.
Also, another naming oddity: they are referred to as the "Mario Bros.", which implies that it is their last name that is Mario... and indeed, their official full names are "Mario Mario" and "Luigi Mario".