I am implying however that you could never confuse a person of an Asian race with a person from, say, an African race.
Of course you could. Without looking up the source of the pictures - which one's which?
People tend to share more genes with people who are more closely related to them - this much is obvious. But different traits have different distributions. Take a look at the maps on
this page showing the distribution of the different alleles for blood groups around the world. It's a clear, regional distribution showing taht people are more likely to be similar to people near them than not. But it doesn't match up to any clear division of races. It's interesting to note, for example, that people from around the Ivory Coast, in the west of Africa, seem to have a very similar distribution of alleles to people from northern Siberia. Siberians don't look like Ivorians, and in lots of other genetic markers they'd be very different, but in this one they're similar.
There was a programme made for Brazillian TV, in which a bunch of celebrities agreed to have their DNA tested for markers of European, African and Amerindian ancestry. Brazil's a melting pot, so almost everyone's got a bit of each. One of the subjects was this singer, Neguiho de Beija Flor:
The guy's clearly got African ancestry, as we can tell from his skin colour and the shape of his face. Thing is, the estimate arrived at by the DNA analysis was that his ancestry was about 67% European. The obvious, external traits we think about when we think of race only account for a part of our genetic makeup, and the actual picture is much more complicated than any usual racial classification.