The fur symbolizes Ms. Brill's inner view of herself. It is fragile, hidden away, untouched, and unique. It is also different, and a bit unapproachable. She is quite fond of her fur, and she refers to it in a way that almost reflects pity. She refers to it as "dear little thing". When she takes it out, she shakes it, gives it a good brush, and takes care of it as if she were taking care of a puppy, or something fragile and defenseless. She protects it dearly and she ensures that it comes out of its box every once in a while.
So much emphasis is placed in the treatment of the fur, that Mansfield basically wanted to demonstrate that the fur and Miss Brill were extremely similar in their lack of approach, their isolation, their loneliness, and their fragility. Once proud of her fur, when the teenagers laughed at her at the Jardins, she took it as a personal criticism. She became sullen and depressed, and she felt ashamed. This is how Mansfield explained the function of the fur and its influence in Brill's personality.