I'm going to answer mostly in terms of US copyright law. As you noted, it does vary--but perhaps less than you'd expect. There have been efforts to keep copyrights similar between countries for a long time, starting from the Berne Convention of 1886. There have been quite a few more international treaties on copyrights since then, and most of the "civilized" world has signed them.
The person who took the picture/video normally owns the copyright.
There are exceptions, such as when/if it was done as a "work for hire", in which case whoever hired them is considered the author of the work and therefore the owner of the copyright.
There may be more than one copyright in play though. If your car has a unique design, there may be at least some room for argument that you own a copyright on it. Under the Berne convention, copyright is allowed to cover: "every production in the literary, scientific and artistic domain, whatever may be the mode or form of its expression." If your car has a basic, factory paint job you probably can't claim copyright on it. If it has a unique and recognizable design, however, you probably can. Even if it's factory paint, you may be able to claim copyright on the lighting (if you've added any).
To give one example of that, the Société d’Exploitation de la Tour Eiffel (which manages the Eiffel Tower) claims copyright on the lighting of the Eiffel Tower, so most night-time pictures of the Eiffel tower are protected. They can't do much about your simply taking a picture, and it would probably be impractical to try to stop people from sharing such photos with their friends--but if you try to sell such photos on a stock photo site, they may well demand royalties.
So, getting back to your question, even if they own the copyright on the video, you may be able to claim copyright on the design of your car. Assuming you can, you may be able to prevent them from using or distributing that video (or at least the part of it that includes your car).
Copyright is a kind of a negative right though. Holding a copyright on your car's design doesn't give you a right to use their video. You have a right to prevent them from infringing the copyright on your car's design, so they can't distribute the video. They have a right to prevent you from infringing their copyright on the video, so you can't use their video without permission.
In a case like this, it's usually to both parties benefit to come to some agreement allowing both you and them to use the video--but that's pretty much outside the law.
Well, there are (probably) laws covering it too, but that would be contract law, not copyright law. Contract law varies much more widely than copyright law, so it's almost impossible to guess at how it might limit such an agreement without getting into a lot more detail about the jurisdiction.
In comments, questions have arisen about whether non-commercial use would still constitute infringement of a copyright.
US copyright law has a provision explicitly recognizing "fair use", so that (for example) a review of a book can contain some limited quotes from that book without its being considered copyright infringement. Most other countries also recognize a fair use doctrine, but don't have any explicit statements to that effect on their copyright laws--i.e., in most countries, fair use is purely common law, not statutory.
The mere fact that a use is not commercial, however, does not necessarily mean that it falls within fair use. For one example, consider Society of the Holy Transfiguration Monastery, Inc., v. Archbishop Gregory of Denver, Colorado. In this case, Archbishop Gregory posted some copied material to a web site. The intent was purely educational and non-commercial.
Nonetheless, the first circuit court ruled that this was copyright infringement, and did not fall within fair use, despite recognizing that there was no intent of commercial exploitation.
The case was appealed, and the appeals court upheld the lower court's judgement.
A long string of comments has resulted in people talking about trademarks and whether trademark infringement could be involved. It was purely a straw man argument, but since it was brought up, perhaps it would be worth writing a little about the basic intent of trademarks, and why they're utterly irrelevant to the question at hand.
The basic intent of trademarks is consumer protection. Let me give a photographically-oriented example. Nikon owns trademarks on names like "Nikon" and "Nikkor". Trademark protection is what prevents me from doing something like buying up some lenses with Nikon mounts from (say) Cosina, slapping a Nikon nameplate on them, and using that to convince people that what they're buying is a real Nikon lens (though as the trademark owner, if Nikon chooses to buy lenses from Cosina and sell then under their own name, that's entirely legal).
So, to repeat: it's about consumer protection. If you tried to sell your Dodge as a Mercedes by Photoshopping a Mercedes hood ornament onto it, that might qualify as trademark infringement.
Looking at the instance Michael Clark cited of taking a picture of the TransAmerica building, we'd get pretty similar results. If I simply take a picture, that might infringe a copyright, but won't infringe their trademark). If, however, I start to sell insurance (from some other carrier) and use my picture of the TransAmerica tower in the letter head, that would probably be considered trademark infringement, because a reasonable person looking at my letterhead could be led to believe that the insurance I was selling was from TransAmerica.
Of course, it doesn't always have to be specifically about selling per se. For one obvious example, if you were deceived about the brand of a car you were renting, that could be trademark infringement as well.
Copyright and Trademark are separate and mostly unrelated. Trademark infringement is primarily about a consumer being deceived into believing that they're getting something different that what they really are. There's nothing in the original question here to indicate that trademarks would be involved in any way.
Usual disclaimers apply: I'm not an attorney. None of the preceding should be mistaken for legal advice.