I am rarely asked for ID when traveling for business, but when you are handing them a credit card that says United States of America for Offical Govenment Travel Only, it is pretty obvious that I am an employee. I had person at a DT start to ask and then looked at the credit card and then said "nevermind."
We are required to use the Gov't credit card when on official business. On the other hand, when I have booked for personal travel and had them an AMEX card, once in a while I am asked for ID. Most hotel chains only require a person to be an employee, but once in a while the terms require both to be an employee and to be on official business.
For airfares it is not a matter of ethics, we are prohibited from using government contract airfares for personal travel. For hotels, our ethics trainers always tell us that we may use Government rates at hotels, if the hotel allows it.
YMMV