Usually no special optical trickery is needed. Perspective is determined by camera-to-subject distance, not by the lens itself. A photo booth can avoid the obvious “big nose” look simply by placing the camera far enough away—roughly around normal conversational distance rather than arm’s length.

Even in a compact booth, there is often enough depth for this: the subject sits back, there is some open space in front, and the camera can be recessed farther into the housing. That can provide enough distance to use a normal-to-short-tele framing for a head-and-shoulders shot without severe perspective exaggeration.

So the answer is generally that booths are designed with adequate physical distance, not folded light paths. Also, passport/ID photos are meant to be neutral and recognizable, not especially flattering, so the goal is mainly to avoid obviously distorted perspective rather than create a classic studio portrait look.