The government does, but they are not allowed to store data, guns are not registered like cars are, their are wholes in the background check system like the "gunshow loophole" or buying weapons online from individual dealers.

Cars are much more highly regulated than guns are. Could somebody just run a bunch of people? Sure. People can kill people with nearly anything. Guns make it pretty easy, it is what they were made for.

Cars are primarily made for transportation. And again, are much more tightly regulated.

Add to that, background checks do nothing for guns already out there. Like I and others have said. The US has more guns just out there than any other western country in private hands. That alone increases the likelihood of one falling into the wrong hands, everything else being considered equal.