It's not a matter of classification, but effect: it doesn't stop people from revealing things you don't want revealed, but from reading on your identity as, essentially, a commercial brand.

Depending on the state, even if the state recognizes the right (it's not a federal right, and not all states, IIRC, have any version of it), it also may not protect you at all, since some states only recognize right of publicity for celebrities. (In effect, your identity needs to be a valuable brand before it's protected in some jurisdictions.)