In the US it varies by state, but generally you aren't just allowed to take the deer home with you either (and for the same reason, it's an anti-poaching measure; here though people usually just shoot the deer and then later claim it was hit by a car, and it's not like you can really tell that easily when you're just looking at packages of meat in a freezer from an already-processed deer).

However, a lot of states do give the game warden discretion to allow people to take the deer home after an accident. So basically when you call to report the accident and speak to a law enforcement officer, you tell them you're interested in the meat and they'll evaluate the decision and may give you permission to do so. That is legal.