You're just not allowed to drive that car in the country. The European Union common market and free movement of goods doesn't apply to cars when the tax revenue of governments is at stake. Finland has had a somewhat similar car taxation as Denmark, and here the government breaks EU law at will; when some procedure is ruled illegal by EU courts, the government replaces it with a new wording and the circus starts all over. No official is ever held accountable.

How does police find those who drive a foreign vehicle illegally? Stop a vehicle in foreign plates, ask driver's license; if it is a local national driving, investigate. Police have very much powers in Nordic countries for these cases (they can always stop a car and investigate, there's no American type requirement for a probable cause.)

And of course, there are always people who report suspected tax evaders to the police.

There used to be a sort of a hunt for these cases between Sweden (who has no car sales tax) and Finland. E.g. there was a guy who lived in Haparanda on Swedish side and had a girlfriend in Tornio on the Finnish side, and stayed too many nights at the girlfriend's place. He got a big fine and had to pay back taxes because he should have registered himself as a local resident and imported and registered the car to Finland as well, including paying the car import tax.

I sold my first car to a Norwegian friend. It was a wreck so I sold it for the equivalent of 50 € (this was 1990). The friend then moved back to Oslo. She was asked to pay the equivalent of about 2000 € in car tax to register it in Norway. She gave it for scrap instead, of course.

Leasing is not that expensive in Denmark in comparison to ownership. Also, due to extreme rusting of your car after a few years due to road salt, I find it gives more peace of mind.

If you are a permanent resident, you can't drive a vehicle with foreign number plates. That rules out leasing abroad.

Many people live in Malmö but work in Copenhagen precisely to avoid this restriction.