Is there anything in the hotel's email to indicate that they were given your card number? I have no direct experience, but my guess is that they do not know which card you used, and it was just a standard email which they send to every guest.
Nope, nothing. Just my stay details and booking number. But I guess they'll know which card I used in the system. When I book FHR stays for Marriott properties and use the app to check in, I could see the card which I used to book the hotel. Never saved any credit card details on my Marriott account so the only way they have the card details is through Amex.