A lady/une dame is a person that you do not feel comfortable referring to as this woman/cette femme. She is likely to look as if she is over forty and probably much older. She is well dressed and looks (very) affluent and (very) self confident, or maybe even self-important. She might even have acolytes. You wonder whether she is a countess or maybe a renowned actress or opera singer (or a famous chef). And then it turns out that she is just the Mayor's wife!
If you work in a service industry or in Customer Service then you probably feel comfortable using "lady" much more frequently than if you do not.
If you were asking a colleague to look after a customer/client for you, wouldn't you use lady/gentleman in front of them even if you thought that they were actually a woman/man?