Ok. Well, what I was thinking for the relationship is that our characters were likely to have known each other most of their lives; at least, those that don't have moving into town as children in their background.
I'm not trying to push anything on anyone, of course, if this isn't how the others see the relationships. But I was thinking that if they were all born on the same day, and they are from rural Nevada, they were probably either actually born in a hospital in a larger city in the area, or something; it wouldn't be likely to get too much attention, since lots of kids are born every day, but if they lived in the same small town, their friends and families would know about it. Perhaps everyone knows of a snowstorm or something that happened nine months before their birth? Perhaps they were the Snowstorm Kids, or something?
In small towns, everyone knows everyone. Depending on how small we are thinking about, there's probably only one school. There may only be one class for each grade. The characters all probably know each other simply from having grown up together. Most likely, everyone in their class invites everyone else in the class to every birthday party; by the time they left kindergarten, at the latest, they probably all knew of the birthday connection. But if all their parents know each other, it seems likely that they would have known even earlier; they probably went on playdates and such all the time.
As they grew up, of course things changed. Everyone tends to either grow out of old playmates, or just grow apart. They don't have to be friends, but they might still have that 'connection.'
My character, so far, is a 'lonely artist' concept; I have him as being a bit of a loner, due to his circumstances, but an easy friend. He isn't much of a fighter, but he is physically fit. His parents put him into gymnastics classes, and he is extremely musically inclined. If this sort of person would appeal to anyone else's characters, I don't mind having him be friends. Otherwise, they could just be acquaintances with this birthday connection. Or possibly they all have this 'Blizzard Kid' (or whatever) connection. What does anyone else think?