No, I do not think we can accurately call Miss Brill's fur a character, but it is a symbol.  A symbol is something that has both literal and figurative meaning: there is actually a fox fur that exists on a literal, physical, level, but it also represents Miss Brill in a figurative way.   

She had taken it out of its box that afternoon, shaken out the moth-powder, given it a good brush, and rubbed the life back into the dim little eyes.  "What has been happening to me?" said the sad little eyes.

It lives in a little box, just like Miss Brill does.  Like her, it seems a bit dusty and old, and she attributes a feeling of confusion to it.  Likewise, Miss Brill will be a bit confused herself as she sits in the park, thinking that she is playing a role in a play as is everyone else she sees.  

Later, when Miss Brill sees the young man and woman who make fun of her, the girl actually says, "It's her fu-ur which is so funny."  The boy makes fun of Miss Brill herself, but the girl makes fun of the little fox fur that Miss Brill thought was so attractive.  Neither the fox fur, nor her appearance is as attractive as Miss Brill thinks:  She does not understand that she is dusty and funny-looking, and she does not understand that her fur has the same effect.  

In the end, it cannot be the fur that is crying, so it must be Miss Brill herself.  Just as she attributed a roguish sort of feeling to it in the beginning, she now seems to attribute a great sadness to it, the sadness she seems to feel.  

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?"

I think the fur represents Miss Brill herself. She is isolated from life, just as the fur. She keeps it stored in a little box, away from everything else, just as she lives in a dingy apartment, away from everyone else. She takes the fur out, and strokes it, and calls it "little rogue" when she plans to go to the park. Stylistically, it is almost as if the author is personifying the fur. This is the only time the fur comes out of its box, and this is the only time that Miss Brill goes out into society.

As she sits on the park bench with the fur, that she calls "dear", she imagines herself a character in a play, but it is really her life playing out. She imagines that she is important, and that if she did not appear on her park bench every week, people would notice. Then the rude young couple make fun of her and her fur. It totally crushes her because their mean comments force her to face the reality of her life, which is very lonely.

When she returns to her apartment, she is downcast, and she puts the fur (herself) back in its box to be stored away. She says she thought she heard it cry because she is crying, in her heart. This again gives a sense of life to the fur, because it is supposed to represent Miss Brill's life.

Miss Brill's fur is an extension of herself.  If we think about it, we can see some striking parallels.  For example, Miss Brill lives in a "cupboard" of a room.  The fur lives in a box.  Miss Brill comes alive each Sunday when she goes to the park.  This is the only time that the "fur" comes alive as she takes it out weekly when she goes to the park.  The fur looks at her with those eyes and asks about why it is here, and what is happening.  Miss Brill never asks those questions of her own life, but very well should.  The fur is mistaken about its own pride and sense of existence.  If we are to extrapolate the idea that it has its own thoughts, then it, too, believes that Sundays are the best day of the week, that it is an essential part of the drama unfolding in the park, that it brings a source of collectivity and unity with its presence.  It believes this because Miss Brill believes it.  When the young couple deride Miss Brill and her fur, she perceives it to be because of the latter, which is why, when she puts it away, she hears it crying.  The fur has become an extension of her, and all of its actions (asking questions, wondering purpose) are what Miss Brill should be doing.  This is why it is so important when it cries.  Miss Brill should be doing that.

It is interesting to see that an object holds that much meaning in this particular work. It is evident that Miss Brill places much importance on the fur.  She seems to dote on it as a child and link her own life to it.  In this vein, Mansfield seems to be suggesting that since we give objects a life of their own, perhaps we can extend this out and understand what they can be telling us about ourselves.