What function does Miss Brill's fur play in the story and can it be called a character?

Miss Brill's fur necklet functions as a characterization device. Since Miss Brill owns the necklet, we know something about her social class, her tastes, and her habits. Her living situation, her one room ("little dark room--her room like a cupboard") tells us abut her economic class, and this can be compared to her tastes to reveal her original social class. Only the monetarily well-to-do can afford the luxury of a fur, even if only a necklet, thus it seems that at one time Miss Brill had a more prosperous life that accommodated a fur necklet.

Her manner of thinking about the fur tells us also about her personality.

Little rogue! Yes, she really felt like that about it. Little rogue biting its tail just by her left ear. She could have taken it off and laid it on her lap and stroked it.

It reveals she is pleasant and good natured and content even though her life is constrained. It also tells us she is careful and thinks well of her belongings, represented by her necklet and her red silk eiderdown comforter. This attitude of thinking well of her things carries over to her with the inference that she also thinks well of herself: she has managed well despite probable reversals, enjoys her few luxuries (like the concert and her "slice of honey-cake at the baker's") and is happy.

Miss Brill does treat her necklet as a friend,

Miss Brill put up her hand and touched her fur. Dear little thing! It was nice to feel it again.

thus as a minor character, particularly as the story concludes with her feeling that her necklet is crying. All these factors combine to result in the crushing blow Miss Brill feels from the callousness of the "hero and heroine" in the park: her image of managing well and her feeling of comfort and pleasure in her little furry necklet are turned to dust and delusion by the view into the young people's perception of her since they see her as aged, pathetic, and out of date.

"Because of that stupid old thing at the end there?" asked the boy. "Why does she come here at all--who wants her? Why doesn't she keep her silly old mug at home?"