The requirement for the driver to "give his name and address and also the name and address of the owner and the identification marks of the vehicle" used to not apply to car parks, but now it does.
This duty is found in section 170 of the Road Traffic Act 1988. The text originally referred to accidents which were
owing to the presence of a motor vehicle on a road
but has been amended to those which are
owing to the presence of a mechanically propelled vehicle on a road or other public place.
The insertion of "or other public place" was effected by the Motor Vehicles (Compulsory Insurance) Regulations 2000, motivated in part by a court decision that a car park was not a road within the meaning of the Act. That case, Clarke v Kato and others [1998] UKHL 36, was about the requirement to have insurance rather than to exchange information about it, but it turns on the same interpretation of "road". Lord Clyde's judgment declined to interpret "road" as including a car park, considering that even though you can drive a car through parts of a car park, it is overall different enough from an ordinary road that it would be inappropriate for a court to extend the statute. It would also not be coherent with the overall use of the term "road" in legislation.
As the presence of this statutory rule suggests, there is otherwise no general requirement for somebody who causes damage or injury to identify themselves. However, their overall conduct could be relevant to the course of action of any subsequent litigation. Giving false details could also give rise to an action for deceit, if for example the other party spends time and money pursuing the wrong person, but that is different from refusing to give them at all.
Another example of an authorizing statute (though a bit arcane!) is section 31 of the Game Act 1831, which gives landowners and gamekeepers the power to ask for the "Christian name, surname, and place of abode" of anyone found on the land "in search or pursuit of game". This would be redundant if there were already a legal duty to identify oneself when accused of trespass to land - a different tort but the same idea.
In the criminal context, Rice v Connolly [1966] 2 QB 414 is a classic case establishing that although people may have a general ethical obligation to help the police, they have no legal duty to give their name or other details - unless there is some specific law requiring it. This was recently applied in Neale v DPP [2021] EWHC 658 (Admin), a case about coronavirus restrictions. Neale's conviction for "wilfully obstructing a constable in the execution of his duty", owing to his refusal to give his details when an officer tried to issue him a Fixed Penalty Notice, was quashed. "Wilful obstruction" requires that the conduct be intentional, and done without lawful excuse (Rice), but Neale did have a lawful excuse, namely the right to remain silent. Although the exact interaction of the right against criminal self-incrimination with civil law is complex, the point here is consistent with the general idea that people do not have to give their names when somebody else thinks they have done wrong.
