Communication between countries are not really good, so yes, if you register your car anywhere, you have to tell your old country to remove their car from their registry. Otherwise you'd have to pay the taxes and insurance in both countries, which you probably don't want to do, right?
But this would also mean it is entirelly possible to have a car registered in two countries. This won't really help you in any way tough. Let's imagine you have a car that is registered in both Hungary and the UK:
- In the UK, road tax and (3rd party) insurance is expensive, in Hungary it's comparatively really really cheap (for me for example it's around 5% of the UK cost).
- In the UK your insurance company will only insure you as the possible driver, so your car won't be driveable by anyone else (unless they also have their own insurance), in Hungary the insurance actually covers the car, so it doesn't matter who drives it.
- In the UK you usually only get a handful of days worth of coverage (usually 90) if you go to other EU countries, in Hungary your insurance covers you completely in the EU for any amount of days.
So based on this list it would be really-really beneficial as a UK resident to drive your car there with the Hungarian tax and insurance right?.
Yes, but it won't work, as your car could only show only one licence plate. In the UK, you would probably use your UK licence plates, as otherwise you'll be driving it there illegally. But if you show the UK licence plate while driving and you happen to be in an accident you'll have to be insured against that particular licence plate, as UK insurers will only insure cars with a UK licence pate, and Hungarian insurers would only insure a car that is driven with a Hungarian licence plate.
Even if you are not in an accident and you are just pulled over by the police you would get fined in either case. If you run with the UK licence plates, it would be uninsured. If you run with the Hungarian one you would be driving it illegally (they would consider it uninsured driving as well)
The only case I can think where this scheme would work is, if you are driving a lot around in Europe, and your home insurance company is giving you a full EU cover really expensively. In this case it's possible that you could replace your licence plates at the border and drive your car in the EU with the cheap insurance, while you drive it in your resident country with the local insurance.
You'd still have to pay both countries' road tax and insurance so this would really-really only give you a slight advantage if you drive around in the EU a lot. And considering this is actually illegal (you can only be resident in one country, and you can only own a street-legal car in the country you are resident in), I don't think this slight advantage would be worth it: if you are driving around in Europe a lot it's much more beneficial to be resident in the country where the car costs are the cheapest anyway.