First, what could lead you to believe that Vulcan does not have a royal family, or multiple royal families? America isn't a monoarchy and we still have the Hawaiian royal family, which include princesses. Also some of our Native American tribes have royal family lines.
Second, nothing says that Vulcan isn't actually a monoarchy. After the Central Command governmental system was brought down in the 22nd century, what would eventually become the Vulcan government is unclear. We really have no idea what they have in the 23rd and 24th centuries.
Third, Spock himself said that his half-brother was the son of a "Vulcan Princess."
Maybe because Star Trek V - the only
movie that TPTB prefer not to acknowledge as truly canon - has the only
mention of a monarchy on Vulcan?
There may have been monarchies in pre-Surak times. I'd have to re-read, but I think there may have been such mentions in Diane Duane's excellent novel Spock's World
But as much as I could wish it, Diane Duane's novels are not canon.