It's wrong in this instance because the woman is perfectly capable of carrying her own purse. The only time a woman should ask a man to carry her purse is when she needs to have her hands free for the short term to inspect an item, as in the fabric of curtains, in a store. Otherwise, she should carry it herself. It's her purse.
The woman in the picture is telling a man to do something which is humiliating. I'd quit that job if I were him.