Well, I can say that you are definitely not alone, because this could be me! I've never felt like I belonged anywhere. I have a few great friends, which makes life more bearable, but it would be nice to feel as if I actually belong on this planet, like I wasn't accidentally misplaced or something. If anyone here is familiar with "The Hitchhiker's Guide to the Galaxy," by Douglas Adams, then they'll understand what I mean when I say that I feel a bit like the character, Ford Prefect: stuck on this stupid world with its ridiculous humans, wishing I could hitch a lift out of here. It's not really a matter of finding a purpose; I'm 54, if I haven't figured it out by now, I probably never will. Or maybe that IS the problem and I just don't know it. All I know for sure is that I have never liked being around people, not even as a child, and I still don't. (It's unfortunate, really, because I always end up in customer service jobs, dealing with people all day.)
In a group of people, I feel like I don't belong, like I have nothing in common with anybody, and I can't make myself be interested in the same things they are. I try, though. I went to a political rally/meet-and-greet type thing for a local guy who's running for office, and felt absolutely NO connection to anyone there. I simply occupied a space, nothing more. It's the same at any event. I'm on the outside, looking in, watching other people having fun and doing interesting things. I keep wondering where I DO belong.
Oh, and I have two cats as well, Elvira and Stormy.