The major theme of this story is betrayal and deception. The story is meant to illustrate the idea that "turnabout is fair play" when we start to deceive and betray one another.
Mrs. Bixby has been carrying on an affair for years behind her husband's back. She has been betraying him. Now she has gotten this mink coat from her lover. She has to deceive her husband in some way that will allow her to keep the coat without his knowing that she has been having this affair. But Mrs. Bixby has the tables turned on her and is deceived by her husband, who gives the coat to his assistant instead.
Mrs. Bixby has been deceived and betrayed by her husband just as she is trying to deceive and betray him. She has gotten a "taste of her own medicine." This is the major theme of the story.