I've heard of no place in the US where you are required to have ID in your possession simply to be in public. You certainly need one when driving a vehicle for which a license is required and when participating in age-related activities like drinking alcohol.
I have no doubt that a policeman would give a person considerable grief if that person didn't supply the ID the policeman demanded, but is it actually illegal not to have one? That sounds a bit too much like Germany, 1938.
Does anyone have a citation of a law or ordinance in the US that requires ID to be in public?