Re: Bigger issues first
The system in the UK is not as you seem to believe.
The 'disc', which every vehicle must display, is the 'tax disc' which indicates that the owner of the vehicle has paid the Vehicle Excise Duty to the government. That disc is valid for 6 or 12 months and is regarded as 'belonging to' the vehicle. It stays with the vehicle and remains valid if the vehicle is sold. (An owner can surrender the tax disc to the authorities in exchange for a pro rata refund.) Without this tax disc, a vehicle cannot be used on the public road.
Insurance to drive a vehicle is a separate matter, bought from commercial companies, and is personal to the driver. Most personal insurance specifies the registration number of the vehicle that can be driven (the registration number is unique to the vehicle and stays with the vehicle and is displayed on the vehicle). A driver must have insurance to drive on the public road and is issued with a certificate which may need to be presented to a police officer if asked for. A driver must also have passed a driving test and hold a driving license (valid until the age of 70) for the class of vehicle being driven.
In fact, the entire set of records - tax disc, insurance, driving license, MoT certificate (vehicle roadworthiness test) and vehicle registration number and ownership records - are held in a central database that can be accessed by any police officer and various other government agencies.
If the police pull you over for any reason, all you need to do is convince them of your identity, then they put a call over the radio to check everything you're entitled to and can also check if the vehicle has the appropriate permissions to be on the road.