From "My Geisha":

...at 7 years of age, she became an apprentice. In the coldest winters, she would practice her musical instruments. As her fingers became too numb to strike, she would dip her hands, chapped and bleeding, into a pan of ice water. This is called kangeiko, or winter practice, and is not intended for cruelty, but discipline. It has been estimated, that the hours required to study and work, to become a true geisha, equal that of a doctor.

The geisha whom I describe speaks French, and English, and Chinese. She plays excellent golf, and tennis, and is a ski champion. She dances the ancient dances and the tango equally well. She can converse with almost any man on his profession. Should he be a stockbroker, she will know the closing prices of the franc in Geneva and the pound in London. She has been bred to please and represents a flawless combination of womanly grace and skills... They are the ultimate we believe, in womankind.

Not a geisha AU so much as: just like there are people who confuse the idea of geisha with simple courtesans, people tend to have the wrong idea about who and what Natasha is. Her reality is so much more complex.
(Bonus points for Clint having this observation.)