In this particular setting and culture, having a name associates a person with a clan or family (the same thing?). The Kitsurugi and Ketsurui clans are highly influential, and having their name is likely to open doors and create opportunities.
Unlike western culture where sharing a surname doesn't necessarily mean you know or are even related to the person (Think of all the Smiths out there), extended families on Yamatai are more likely to do favors for each other if they share a name. It's nepotism, name-based favoritism.
The second paragraph of this explanation is supposition on my part, using what I have observed. But it makes sense. The first paragraph is a definite, though, just look at how many of Yui's daughters are Taishos...
Which, now that I think about it, that makes less sense for why nekos would have random last names (even if they are grouped). The real families might be perturbed at finding they're unknowingly sharing their surname, wouldn't they?