(They do have other names, supposedly surnames or patronymics or clan identifiers or perhaps something alien, but those are apparently seldom used in official context, given that we have never heard a single one!)
Not quite true. In Star Trek III, Sarek is referred to as "Sarek, child of Skon, child of Solkar". In English, at any rate, it seems they make the distinction by patronymics. My guess would be that a name continues to add patronymics backward until there is no confusion or until one reaches the name of a Vulcan no one will mistake for anyone else. Kinda like the way we refer to locations:
"New York" - 'nuff said. Everyone knows this city.
"Buffalo, New York" - There are other Buffalos, so we add the state, New York. Everyone knows that state. "nuff said.
"Valdosta, Georgia, U.S.A." - Who the heck knows where Valdosta is? So we add the Georgia part. But Valdosta sounds like it could be Russian. So we clarify that we don't mean the Republic of Georgia by adding U.S.A. 'nuff said.
Spock, for example, can add until he has made which Spock he is clear - "Spock, child of Sarek, child of Skon..." but they get to stop at Solkar, because he was one of the three Vulcan signatories to the Federation charter. 'nuff said. (For that matter, at some point I expect Spock's renown was such that he could just say "Spock".
Other Spocks would have to clarify.

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