Not contend with simply filling in Jack, bureaucrats would simply come up with a last name for them. Or - if they felt so inclined - ask the person to come up with one themselves. They would often chose their occupation. Americans know that a lot of modern family name spellings in the US are the result of careless bureaucrats at Ellis Island.
Back then, family names, house names, last names and surnames weren't necessarily the same thing. One might have more than one. You may be of one House, but your last name was Jackson (son of Jack). Also somewhat inconvenient for these bureaucrats with few fields on their forms. And also, stop changing last name from generation to generation.
Though, fortunately for these bureaucrats, unlike modern programmers, when the map did not fit the terrain, they could simply alter the terrain.