Manchester used to have the most secure, long-term parking for cars in the UK. They used a hangar on the airport itself, and you drove into it and were taxied back to your terminal. But then they built the new terminal, and that was that.
I can only imagine the car owner has become confused because, as you say, leaving keys inside any car is asking for trouble. It's odd they believe they can't claim against the company, though; any company looking after your vehicle for two weeks has a 'duty of care' in law, so I'd be interested to read the booking Ts and Cs. Even if they say something like 'we take no responsibility for your car' I would suspect that could be challenged under the Unfair Contract Terms Act 1977, because it sounds very far from reasonable to me.