Coinbase is a digital asset exchange company headquartered in San Francisco, California. They broker exchanges of Bitcoin, Ethereum, Litecoin and other digital assets with fiat currencies in 32 countries, and bitcoin transactions and storage in 190 countries worldwide.
American Express is an American multinational financial services corporation with operations in more than 130 countries. Amex cards are accepted in most places that accept credit cards.
The short answer to the question is no, Coinbase does not accept American Express. The longer answer is that while Coinbase used to accept Amex cards, they stopped doing so in early 2018.
The reason for this is that Amex has stricter rules and regulations around merchant processing than other credit card companies. This made it difficult for Coinbase to comply with all the rules and regulations, so they decided to stop accepting Amex cards.