Most(all?) types of intelligence are positively correlated. So if you have a natural ability at one of those things, then you're more likely to be high in the others too. Even something as simple as reaction time is correlated with IQ.
Most engineers I know have trouble reading facial cues when interacting with other people. Also they don't realize when they are being manipulated and they are incapable of manipulating others to get what they want.
People who have high emotional intelligence can essentially read the other person's emotions while they're talking to them and they can use the feedback to adjust their speech to make it more convincing. I think that's definitely a form of intelligence and it requires a lot of practice as well as natural skill.
I've met some very uneducated people with very high emotional intelligence.