Why use the verb wear for makeup like clothes?

Since makeup is a noun (a thing), and just like clothes you are putting it on your body, it makes sense that you would use a verb with it. You are using it as a predicate noun, so you have to have a verb to complete the predicate.

Since you are talking about the stuff called makeup, you would need a verb with it. You can use it in the verb form, but that would refer to the act of making up (compensating for) something, like making up a test.

Originally, the noun makeup probably came from the verb form. You would wear it if you wanted to make up for uneven skin, wrinkles, or blemishes. One of the major makeup brands tends to imply this in the name -- CoverGirl, though they might have meant a "cover girl" as in a model. But that is what their product does if you think about it, it covers a girl.