I also have an extremely rare surname. Back when the internet was still pretty young -- about 11 years ago, I think -- I was contacted by email by someone from Brazil who'd seen my name in an industry news item, and wondered if we might be related. We compared the information we had about our families, and it seemed likely that his grandfather and my grandfather were brothers. His had emigrated to Brazil, while mine had stayed in Central European Country. He mailed me some photographs, and there was a definite resemblance to my father's side of the family. It seems there were eight brothers in the family, a family feud, and everyone dispersed. So far by googling I have turned up a number of people in Germany with the name. I'm guessing they are probably distant relations, but have never tried to contact anyone.
When I joined FB, I searched for my surname -- I only came up with one person, and emailed to say "It's a very unusual surname, I wonder if we might be related. Here's some of my family history." She didn't have anything going back that far, unfortunately, so we still don't know. My guess is that my grandfather and her great-grandfather might have been brothers, or maybe cousins.
In the end, though it's vaguely interesting to discover distant relatives and piece together family history, I found there isn't much more to it than that. My "cousin" in Brazil likes to keep in touch, but we don't really have anything in common. The girl in Germany is an FB "friend", but we don't actually have anything to say to each other. Ultimately, you're probably no more likely to want to know this person who has the same surname as you than any other random person. But that isn't a reason not to have a go anyway!