Depends on the country. In the US, the scanners don't even record more than the last 4 digits, CVV, and expiration once the transaction is approved. Elsewhere, you may see the old paper forms which take an impression of the card (of if the numbers are not raised, are hand-written).
All of these systems carry a risk and it can be heightened in some countries. But, the better question to ask is what you would do if the clerk had simply said, "no problem, when you have completed the form, we'll complete your registration and issue your room key" ???