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It is either that:

a) they don't have any hats, i.e. that they are lying about having any hats at all.

b) they have some number of hats (n >= 1) and at least one is not green, because they are lying about the color of all their hats.

I would be happy to learn how I am wrong.

I wonder what color is a hat that doesn’t exist.

If a person doesn’t have any hats, is the statement ”All my hats are hats” false?

That is a tautology. A tautology is always true.

I think the question is more about liars than it is about hats.