It's the latter, but i'm seeing it more like a kind of astral projection. The mind can create a second body, enter it and leave the original one behind, which seems to sleep for onlookers.
The second body can feel pain and if it receives mortal wounds, the person awakes in his/her normal body.
Sounds to me like you are operating a drone. Interesting that your hero can feel pain. Presumably, this is not an invention then, it's a biological ability. It's an interesting idea for a hero though. It brings other questions--if the hero is killed, what happens to the hero body? Does it disappear? Can the hero create a third body/another drone? If so, can he create multiple bodies at the same time?
I think in the case of the drone, it would all depend on how the body is created. Does it split off from the original person? Or can this hero make the body appear anywhere he wants? If there is trouble say 500 miles away, can the body instantly appear at the trouble site?
If so, then the hero can operate from a safe bunker, where it would be very difficult to know. What happens to the body when trouble is over? Same thing as death. Does it merge with the hero? Does it disappear?
The character would have to be fleshed out a bit to really decide how difficult it would be.
Humans have certain habits... patterns... unconscious as they may be. The created body might appear random but, most likely, there will be something common between them. Face shape.. eye shape... something. Perhaps it's unconsciously choosing the same clothes. OOOH! Clothes! Unconsciously choosing the same wardrobe as the secret identity. Unless the super identity wears a uniform.
Voice? accent? Gestures and mannerisms? I was amazed at how quickly I could recognize my father-in-law when he plays hockey. He might be in his padding and gear but he skates the same way he walks. When you are with someone long enough, you begin to recognize them by body language.
What about smell? Body smell? Aroma? Could a dog tell the created body is the same as the original body by sense of smell?
That may be provable, but in theory, someone would have to suspect that there even IS a secret identity. It's safe to say that if the originator looks completely different, (maybe even a different race), why would anyone suspect the originator?
Think about it from the view of the average citizen. Superman comes on the scene. Why would anyone even suspect he is living as Clark Kent, or any human? Heroes with masks? Sure. But heroes like Superman and Wonder Woman don't hide their faces.
Now factor in a better disguise than a pair of glasses.