Romans had nomen and cognomen, which were inherited names like surnames. Greeks and Jews, like contemporary Icelanders, just had patronymics: John son of Zebedee. (See also the list of high priests of Israel.)
Less often, they had nicknames indicating jobs or characteristics: Simon the Zealot, Judas of Kerioth, Jesus the Nazarene. In narratives, those distinctions would only have been used when felt necessary.