How come I don't feel what other people feel? Is something wrong with me?

There is nothing wrong with you, you just happen to handle things differently. Some people like to talk things out and express how they feel and seek comfort and support while you perhaps aren't necessarily as bothered by things that may hurt other people. I think it's good that some people can handle their own problems. Personally, when I'm visiting a friend I'd rather talk and enjoy ourselves and have a good time than cry over drama or vent out stress and anxiety.