The fur Miss Brill wears is the most loaded symbol of the short story. On one level, it represents Miss Brill's individuality and sense of pride. She loves the little fur, and she wears it when she especially feels like looking her best. She is excited when she takes it out of its box.
The fur is also evocative of her desire to have a friend. Miss Brill lives a lonely life. Deep down, she seeks connection with other people. By focusing on details such as the fur's eyes and referring to it as "[d]ear little thing" or "little rogue" she is ascribing a personality to this object. That she refers to the fur as a "rogue" is telling. Like a rogue, Miss Brill lives on the edges of society with neither family nor friends. When she puts it away at the end of the story, she even thinks she hears it crying. However, this could be Miss Brill herself weeping, and projecting her sorrow onto the fur.
When the young couple insult Miss Brill's fur, they are doing more to her than saying she has poor taste in fashion. They are unknowingly insulting her sense of self and making her feel more alone. When Miss Brill weeps and puts the fur away at the end of the story, she is symbolically showing that she now feels deep shame about who she is. Her fantasized connection with the other people in the park is revealed as nothing more than a comforting illusion.