If it's like the UK system, then yes the police could easily spot the discrepancy in the make, model or colour of the car. The DVLA has all the data that's on your car log book/ownership document, and the police can access it all through their computer system, which reads the number plates of cars that drives past and flags them up (e.g. as stolen, wanted, lack of insurance, etc). All of what's visible from a police car as you drive past (make, model, colour) needs to match, though I'm sure cops don't check that on every single car that drives past. But if your story requires the car thief to be caught that way, it's totally plausible.
If your story requires the car thieves to escape detection then they have to be very sophisticated about it.
If automated registration plate recognition is a possibility, then all they have to do is nick plates from a car similar to their stolen vehicle.
They'd also have to be sure that the stolen plates have up-to-date tax, MOT and insurance as well. And if they have all that, the car's regularly on the road. So it won't be long before the driver notices that someone's stolen their number plates and reports that to the police. Meaning their computer system's going to have the stolen number plate down and it'll flag the car up for that. If the theives steal plates from a car that's off the road, then the police computer's going to flag it down for not having tax, MOT or insurance. If they steal plates from a car that's off the road that's the right make, model and colour then pay for the tax, MOT and insurance, that might work, but paying for stuff defeats the purpose of thieving.