It's a fairly different thing—emotional intelligence requires not only the ability to assess other people's emotions, but also enough self-awarneess and the ability to not be responsive to one's very own emotions, which is an entirely different set of skills from executive knowledge like intelligence in A or B.
Emotional quotient was the nothingburger you mentioned, it has no basis or operationalization to it whatsoever. It's also usually paired with spiritual quotient, which was also equally bull.
This is starting to become quite a complicated topic, but long story short, the current emotional intelligence and EQ have the same roots in developmental psychology fads from decades and decades ago and are as common in self help fad sciences as they are generally discarded in health sciences for the same reasons. Emotional intelligence is just more pervasive, as it is less descriptive, but both have just about no predictive validity. It's not "wrong" to use as a colloquial word or a figure of speech that it is in basic conversation, but as a defined term it's as valid as others like it (so to be clear: Not at all), such as self esteem and psychopathy. But more accurately, an "emotionally intelligent" person would just be an intelligent and agreeable person, who's (optionally) conscientious enough, if you want to fulfill your personal list of requirements.
That's why I said that it's "kind of a thing" as you can say the word and convey a sort of descriptive, but nebulous meaning behind it, but you can't really define or separate with it, unlike EQ, which tries and fails very hard and visibly. Like, one's a ruler that either shows the obviously wrong number or visibly steals notes from a good one, while the other stretches to show the expected, so both are bad rulers, one just more blatantly so. I know that I'm explaining it quite badly.
