Touch ID login allows you to use your fingerprint instead of entering your Amex User ID and password to log in to your American Express online account. Your User ID and password will be encrypted and stored in your device's Apple iOS Keychain.
In order to use Touch ID, you need to enable it on your device through your iOS Settings under the "Touch ID & Passcode" section.
By enabling Touch ID login, you are allowing anyone who has a fingerprint stored on this device to access your account. So please review the fingerprints stored on your device and make sure all should be authorized to access the account information available through Amex Mobile.
American Express does not have access to your fingerprint information.
For information on how Apple uses and stores your fingerprint and Keychain data, please see Apple's Privacy Policy and iOS Security Guides.