I know that the question “What color eyes do you have” is quite common but why not say “What color do your eyes have”?
TOEIC listening, talks: Sales executive leaves voice message for his co-worker asking him to get in touch with accounting department[YSaerTTEW443543]
From what I’ve googled, you can refer to the color of your eyes in different ways:
1–Based on your personality, what eye color should you really have?
2–What Color Eyes do You Have?
3–What color are your eyes exactly? What’s your natural eye color?
Similarly, hair: What Color Hair Do You Have? Will this quiz finally tell you what hair color you should have?
My wild guess you can only tell about what color eyes\hair\shoes\suit you have (= possess). Saying otherwise would kind of ‘humanize’ those things.
The first question to me suggests that you can change the colour when you feel like it. The second question suggests that your eyes have nothing to do with you.
Maybe I’m being perverse but that’s how they sound to me. The simplest question is - What colour are your eyes?
This is how we, non native speakers, say it.