Have you noticed how many brick and mortar stores ask for your email or phone number at the register during a purchase? They absolutely can tie offline purchases to online activity with that information (not to mention your credit card).
Some can, some can't. Ad networks such as Facebook actually don't let much information escape, getting even basic demographics on who saw one's Facebook ads can be hard, getting identifiable information is, to the best of my knowledge, not available around impressions.