South Africa is a nation and the people who are born there are South Africans. Nationality, ethnicity, and race are three different things. For example, a white person born in South Africa is not Xhosa or Zulu either. In fact, the whole history of apartheid in South Africa was about the fact that the white settlers did not view themselves as Africans.
As for us in the Caribbean, we aren’t African just by skin, but by our culture as well. We have African foods, music, folk stories, and even some African words in our language. But as I pointed out in this article, even though we have a cultural connection to Africa, many in the Caribbean feel a sense of shame over this African identity, so many of us don’t claim it.
Every race may have begun in Africa (that’s debatable) but obviously when those other races left Africa, they developed their own culture and evolved separately from those who remained in Africa. Again, the point here isn’t that we are African simply because we look like the people who live in Africa, but we are connected via a shared history and a shared culture. I got to experience this shared culture firsthand when I was in Ghana.