They should but only 40-50% do, I write "Ask for ID" on the signature portion. The ones that read it, ask me for it. Better than a signature that a cashier won't question anyhow.
So the pedantic response to your second sentence is that they consider the See ID irrelevant (and once there is a signature, it is apparently a violation of the merchant agreement to ask for ID, but I didn't look for that in the operating regulations).