Keep in mind that the idea of everyone having last names is kind of a recent thing, and by no means is it universal. In the medival era, which is what the Zeldaverse is mostly based off of, only the noble classes would go by any kind of surname or family name. The peasant class didn't have last names, but they would probably say "I'm Will the Thatcher" or "Greg the Cooper". Then later on the "the" gets taken out and boom, last name. Other times people may say "Bill from Gloucester" or "Leonardo of Vinci". Yeah, Leonardo did not have a last name. "Davinci" is Italian for "Of Vinci". They didn't feel the need to specify unless there was more than one person with the same given name.
I imagine if Link had to introduce himself and he had to specify he'd say "I'm Link from/of Kokiri Forest/Ordon/Outset/Skyloft/Hyrule or state his title if he has one, like being a blacksmith's apprentice, knight in training, or perhaps Hero of Time?
So that's why last names aren't a big deal in the Zeldaverse.
