I’m not sure if there is a definitive answer on this but let’s see what the evidence says.
They are called the Mario Bothers kind of like how Jacob and Wilhelm are called the Brothers Grimm.
Jacob and Wilhelm’s last names are Grimm because they are both brothers who share that name.
Obviously Mario and Luigi are both brothers, in fact, they are the Mario brothers.
Thus we can conclude that Mario is the last name of both Mario and Luigi.
I find it odd though that Mario’s parents would name him Mario Mario. Hmm…