For the charge cards. You use to be able to goto nearest Amex office and get a replacement. I did it once in the Early 2000s.
Same thing happened to me when I received my Amex Surpass in January and my last name wasn't on the card. Called to get another one and they said I had to wait 10 days, and said I had to call back in 10 days and request a corrected card again. The guy I spoke to initially said it was their policy, but when I called back 10 days later, the rep said it was part of a new law, I think designed to prevent credit fraud. At any rate, I resented having to do it for something that was their mistake. They did overnight the second card though so that was nice.
When my Amex plat was damaged they overnighted me a new one via FedEX. God forbid I couldn't spend on it.