My cousin is going to book a room for me in Istanbul using his Mastercard. I've heard that the hotel may ask me to show the credit card with which the payment was made. If the cost of the hotel is fully payed using the Mastercard, do I still need to show the credit card?
Any hotel will almost certainly ask you to swipe (any) credit card upon check-in for security against other bills you may incur in the hotel (restaurant, mini bar etc). Some may take a debit card or even cash, but best to check with the hotel in question if in doubt.
I have not heard of hotels requiring the same credit card as used in the booking to be shown, but again, check with the hotel (or Booking.com) if in doubt.