For example, when I play as China, it seems that most of my great people are Chinese. Does playing as a specific empire cause you to have great people that come from the empire? Or is it purely coincidental?
No. The list of great people available is an easily edited file split into categories for the artist/writer/scientist/whatever. The name of the person is randomly selected from this list.
I don't know the criteria Firaxis used to choose the people on their list, but there aren't 'great people' on their list who are from many of the default civs, just mainly Western or Oriental countries.
I don't think this is the case, in part because many of the lack of great people for many nations. I don't think there is a single great person from Zulu or Siam, for example. Rather, I think you are experiencing confirmation bias because a large number of Great People in the game are from China. Keep in mind the vast majority of great people are chinese, european, or american. If you look at this list the majority of non-western GPs are chinese.