It's very unlikely that the average and the median IQ values are exactly the same--this is only true for a perfect normal distribution.
Your general point about medians and means is correct, but it's not true that the mean and median are the same only for a normal distribution. Trivial counterexample: if half the population had IQs of 99 and the other half had IQs of 101, the mean and the median would both be 100. You can think up more complex variants yourself.
OK, but the point is that while there are many (in fact an infinite number of ) theoretical circumstances in which the mean and median are the same, this is not true in general, and it shouldn't be assumed to be so.