How common is it to receive letter from admissions asking you to apply, out of the blue?

Well, those two examples are different. It’s likely that BMW targets its ads more than the Italian restaurant does. This is why we get glossy ads for things like cruises (and sadly, more recently, for retirement communities). If you get a personalized ad for a BMW, that probably means that some algorithm somewhere calculated that you probably have enough money to buy one. It’s the same for a promotional letter from Yale. Not all college students get them, so there is some algorithm that says that this student may possibly be a plausible applicant. But it’s still a pretty wide net, so nobody should take it very seriously.