We have no reason to believe there exists a trait like intelligence for emotional/social skills, all we have is people establishing its existence post-facto like you are doing here (from seeing some people are good at multiple social skills and assuming the existence of a common root trait).

It's like seeing some people are on average better at sports than others and saying there must be a "sportiness" trait that determines how good you are at sports. Instead we know it's a combination of various different things, many of which are independent of each other.