Do American Express cards have a similar authentication?

For Amex cards the 'extra level of authentication' will work slightly differently. Amex uses the AVS-Address Verification System.

On the payment page on BookMyShow, you will be asked to enter your billing address. This will be checked against your billing address the bank has on its records.

If the addresses don't match your payment will be rejected. If you don't remember your exact billing address, you can confirm it from your Credit Card statement.