I would like to expose my opinion about this.
In "What colour or color hat do you like?" the word colour-color is not a name but an adjective. Like probably would be colorous, color-filled, or similar, forcing the interpretation. So, "What?" goes with hat, and hat is qualified by the adjetivized noun, therefore an adjective, in our case colour/color. The question would be "What hat (with color) do you like?". The same applies to "size shoe"="shoe (size-wise)", and so on. In a word, what does not inquire about the color or size or similar, which are adjectives, but about the object (hat, shoe, etc.), in order to distinguish or separate it from the rest of them. We do not want to know what color or size is, but what hat or shoe it is, (even though always referring to colors or sizes). I might be wrong, but this is which I had always thought.
Summing up, Elaine Koh's wanted to know how to ask about the colour of the hat, not about what hat it was. So, the correct question would be
What colour/color is the hat that/which Jane bought?
What colour/color is the hat Jane bought?
What is/What's the colour/color of the hat that/which Jane bought?
What is/What's the colour/color of the hat Jane bought?
Also
What colour/color has the hat that/which Jane bought?
What colour/color has the hat Jane bought?