People who think they're better than others for being empathetic, are narcissistic, definitely.

However I do think that if you are easily hurt by the emotions of others, it's not necessarily a sign you lack empathy. It could be that you have gone through trauma from abuse and you have developed an extreme sensitivity to others' emotions, like "walking on eggshells". I know that I personally was ultra-attuned to my mother's emotions, and if I sensed she was feeling negative emotions it would majorly affect me, because I knew that she would abuse me if she felt I was causing her pain. And her main method of abuse was guilt-tripping, "you caused me pain you're a bad person" kind of thing. So yeah I think I was feeling empathy for her, sometimes maybe feeling WORSE than she was because I would not only be sensing her negative emotions, but also feeling my own guilt for feeling as if I was "causing" these negative emotions.

Like imagine you suggested a movie to your mother, and you sit watching it beside her, unable to focus on the movie itself, only on how she was feeling and reacting, because if she's not enjoying herself it's all your fault and you're a bad person. That's the kind of thing that was the norm for me.