Its a tempting hypothesis. And its entertaining to look at the wives of your relatives/close friends and theorize which attribute of their mothers they replicate (likewise for husbands/fathers.) But this seems like a difficult hypothesis to carefully test. Here is one attempt. Assemble a dataset of bi-racial families. We want the race of the father and mother, the sex of the child, and the race of the child’s spouse. To control for the racial proportions in the population, we compare the probability that a bi-racial male with a white mother marries a white wife to the probabiltity that a bi-racial male with a black mother marries a white wife. The hypothesis is that the first is larger than the second.
Now, marriage is a two-sided matching market. This means that we cannot jump to conclusions about the husband’s tastes on the basis of the characteristics of the wife. It could be that this husband would prefer a black wife (other attributes equal) but the best match he could find was with a white wife.
For example, an alternative story which would explain the above statistic is that black spouses are generally preferred but having a white father makes you a more attractive match and so bi-racial children with white fathers are more likely to match with their preferred race. (Any theory would have to explain why there was a difference in the ultimate match between those with white fathers and those with black fathers.) But the data would enable us to potentially rule this out. If this alternative story were true then bi-racial daughters with white fathers would also be more likely to marry black husbands than those with black fathers. That is, girls marrying their mothers rather than their fathers, the opposite of what the original hypothesis would predict.
So if the data showed that boys marry their mothers and girls marry their fathers, we could rule out this particular alternative story. Of course there will always be some identification problem somewhere, and here the following story would be observationally equivalent: having a white father makes you a more attractive mate, women like white men, men like black women. (Allowing men and women to have different racial preferences adds the extra degree of freedom to explain the [hypothetical] data.)