This happens?

What? Having friends, or having friends in different time zones?

a) Yes, it does.

b) In general population, I would guess it doesn't; unless there are specific circumstances, e.g. your relatives moved to a different part of the planet. But here on LessWrong we have an international community, and some people visit meetups in different countries. There are probably only a few who have travelled to a sufficiently distant time zone. But you don't have to be one of those; only to be a friend with one of those.

Friend is a matter of definition

Sure; if we taboo "friends" it means something like "people whom you trust enough to do together X". For different values of X you get different sets of people. (X = "have fun together" or X = "start a conspiracy to overthrow the government")

In my experience, socializing with people at work does not carry over into socializing after work

Similar for me, 2 exceptions in 20 years.

I usually socialize with my neighbors, with people I have or had some hobby in common (such as LessWrong), and sometimes I meet friends of my friends and they become my friends.

Of course with my neighbors the expectations are low: generally just being nice to each other in case someone will need a little help from the other, and to keep communication lines ready in case there will be a shared problem to solve. Saying hello to each other, bringing cookies, sharing a glass of wine once in a few months. With people I found through my hobby I expect to talk about the hobby, and later about other topics; and if the relations are good, maybe even spend some vacation together.

I always figured it is more natural to hang with relatives, perhaps kinship based tribes should be reinvented

I was thinking along similar lines once, but almost everyone from my family lives in a different city than me, so it's not an option. But generally, "relatives" and "friends" are two different categories; I cannot realistically expect my relatives to have similar hobies as I do. With them, it is a different way of spending time; just being together, being a tribe. With friends, it is talking about hobbies, making hobby-related plans, and later also being a kind of a tribe -- though this part is more difficult because "clicking together as friends" is less transitive relation than "being related by blood". There are many situations where I am friends with X and Y, but X and Y do not like each other. (So I cannot convert "having 2 friends" into "having a tribe with 3 members".)