Core to emotional intelligence as construed by Hindu scriptures is "awareness " and "self introspection". Awareness of the fact that the soul or athma as we call it, is transient. Awareness that this life and body is just a garb. The soul transcends the garb. And hence it is dissipative to be attached to the body and the emotions attached with it. Armed with this awareness, we will be equipped to not let the emotions control the intellect. We will learn to not let emotions like love, desire, hatred, anger, jealousy control our thoughts and action. As I have time and again emphasized in my articles, attachment induces a multitude of emotions.Introspection helps us get to the crux of the trigger and address it. Hindu scriptures never talk about assessing or managing others' emotion. It is rather self introspecting. It recommends introspecting on what triggers our own emotions and how can we manage them. Beware that "manage" does not mean suppression, but rather introspection.
For e.g: When we are deeply desirous of a particular goal, we tend to be lopsided. For the desire of reaching the pinnacle of our career, we might de-prioritize our personal life, and lose out on the most important phases of our own life and others. At the same time being attached to our personal life and kin, can be stifling. This is where the awareness and introspection as prescribed by spiritual texts come in handy. Realize the fact that what transpires now is just a passing phase of your soul and it is wasteful to expend our time and energy fondling the emotions of this phase.
This is the mantra of emotional intelligence as construed by Hindu scriptures