I'm sure there is no reason why you couldn't, and it is very common when someone who is insured to drive their own car, then drives another insured car. They will be insured for at least 3rd party personal injury fopr any car in their own right, and possibly as a named or "any driver" for the car they are driving. However I am puzzled why you would want to do this deliberately, since only one person can drive the car at any one time? Surely it must be cheaper to insure the car with addiitonal drivers on the one policy. Most importantly if you had to make a claim it may become very messy because you can only claim on one policy, and each insurance company would want you to claim on the other. I would also be curious as to how you would deal with the issue of who the main user, registered keeper and owner is on each policy form. I imagine an insurance company would ask some pretty searching questions as to why someone wanted to insure another person's car.
The main point is, you can only claim once, and to try to claim for the same loss on more than one policy is a criminal offence.