Actually all that has been said does not explain why English prefers to say
- 1 What colour are his eyes? And not: 2 What colour have his eyes?
I think that the variant 1 was originally
- Of what colour are his eyes?
In the course of time "of" was dropped as describing the colour of something is a frequent topic. So we actually have a shortened expression.
The drop of "of" is for me the only reason why English says: What colour is her hair? and not: What colour has her hair? as a lot of speakers of other languages would say.
It would be fine if I or others could find something that would back up my theory.
Added: In this question LinkLink we find "Every ball has a different color" and "All balls are of different color.