The owner's not identified, the plane is, so that ATC (or, in the case of an incredibly unusual emergency, another plane) knows who to call.

She can always operate her plane in VFR mode in airspace where (because of fewer safety concerns for those modes of operation) a transponder doesn't need to be turned on.

Airspace is a 'public' space, but it's public space that requires very clear tracking of objects occupying it, because of high risks to human life. It requires permission, and following very particular rules and notification for entering it. Flying is not like taking a stroll down to the corner store.

Notice that driving also requires you to broadcast your vehicle's identification, in the form of a license plate. But because the safety concerns in play are different, it's only broadcast in the visible spectrum.