Well, right, you might not immediately understand and identify with someone else's problem. That is, you won't empathize with them. But you could say that you sympathize with them, if their pain causes you to feel pain as well (or joy, or...whatever). Or their situation may cause you to feel compassionate, if their suffering makes you want to help. I think that there are already words that name the feelings you're talking about.

"Empathy" has a generally-accepted meaning. You can disagree with it if you'd like...but it will hinder communication to use words in ways that they aren't commonly used.