Seems like some kind of weird quirk that the government doesn't already have this information readily available. Why isn't there a registration process for the person that leased the car?
The police should have information on people who have broken the law (assuming the laws are reasonable and proportionate – for the moment, let's make that assumption). The police should not have information on non-criminals, except as far as it is genuinely necessary for an investigation. (To the extent that the police do things other than investigating crimes and making arrests, the relevant information should be compartmentalised and handled separately.) I am willing to tolerate large amounts of inefficiency, and even some bad guys getting away, if it ensures that the police do not begin to get results by looking only under the street light (which, if nothing else, will lead to sophisticated offenders getting away more easily). Pre-emptively requesting records just in case they're needed is a very, very bad practice, and we must oppose it if we want to live in a free society.
This is also why I tolerate the widespread use of CCTV cameras, but strongly oppose CCTV networks. Closed-circuit television needs to be closed-circuit, with friction of access requests proportionate to the amount of footage requested, or it goes from an accountability tool to a mass surveillance tool.