How do you go about identifying your feelings?
I used to try to identify my feelings by introspection / phenomenology. I thought that if I focused on the feeling then its name would suddenly occur to me... That wasn't working out so well for me. Then I learned about analytical functionalism which is very similar to the way in which Linehan advocates people can learn about feelings. By looking at triggers (thoughts / perspectives) and looking at actions or action urges.

How do you go in idenfifying feelings in other people?

I think both of those abilities are learned. Our parents vary in their own insights and in their ability to teach us. We can learn this later in life...