I've been thinking about, well, I can't say about Zaun/the Undercity and names but rather about the conversation around that.
This conversation mostly exists in the context of Zaunite characters not having a last name, and very frequently* it centers Viktor's lack of family name as telling of a cultural pattern.
However, I feel like this reading of Viktor's situation in particular overlooks one aspect that i think it's invaluable when discussing Viktor's position in Piltover and Zaun alike: Viktor is an immigrant.
He is the only character in the show with an accent like that. During Season 1 Act 1, we see him try out idioms ("You have to... crank it."), as if they're outside his understanding of the language, but he is fluent enough that he can glean the meaning and incorporate it without needing explanation.
This all points towards a person who came into this society at a young age (obviously, given the flashback) but was not born there.** Children who are born in immigrant communities are exposed to the dominant language in daily life and speak it with whatever accent is prevalent in their immediate surroundings, yet Viktor doesn't sound like any other Zaunite.
And there is much to be said here, about Viktor's sense of belonging, why it would not have been that difficult for him to let go of any community in the Undercity, given that we don't actually see him having one, as well as his lack of understanding of what life for certain populations in Zaun is like. The point I'm getting at however, is that I believe it's less that Viktor doesn't have a last name because he's from the Undercity, it's that Viktor doesn't have a last name because no one in either Zaun or Piltover can pronounce it.
And Viktor, being someone who is perpetually striving towards the top, simply shed it as everything else he thought held him down.
*Granted that impression might be skewed in my case since he's a character who I seem to seek out the most, call me biased.
**This is also why I don't ascribe to the idea that his disability is a product of Piloverian ecocide. I've said it before, but sometimes you're just dealt a bad hand. (The disease very much is however.)