I think that emotional intelligence must be developed as other forms of intelligence. It seems that expectations about how to develop emotional intelligence used to be different for men and women and, for some people, it is still like that.

My grandma used to be able to anticipate whatever my grampa were going to need/want, read his emotions even before he was aware of them and start providing whatever care he needed. But I never felt it was "natural" but the result of a whole life together. That is, it was learned and I think it prioritized learning "male" emotions, since she wasn't as good reading my mom, her daughter.

So I think that you have a point about men having emotional intelligence.

And we all need to learn how to use it better. But not just with the other gender, but with everyone in general. And each persons emotions are different, so it might take time and learning about the person before you can do it properly. But this is just my take.