The glamor that we normally associate with fur is all but missing in "Miss Brill."  But the fur that we hear described none-the-less evokes a glamorous past.  The key word there, though, is "past."  Miss Brill is no longer young.

The fur to which she she clings,  evokes feelings of youth and vitality for her.

Notice how her language becomes so affectionate  and emphatic when she thinks about the fur.  "Dear little thing!"  It has a significance (and even personality) for Miss Brill which belies the somewhat shabby appearance that the park-goers and readers see.  One might even argue that the fur is Miss Brill´s alter ego, which is perhaps why she takes it out of its box and dusts it off for this trip to the park, even though the weather doesn´t really call for it.

Among the other park-goers Miss Brill is at home.  It is there that she manages to find a meaning that is otherwise missing.

I would suggest that the fur becomes a synecdoche for the youth and vigor of Miss Brill.  One might almost argue that the fur is representative of her sexuality (notice that it goes back in the box at the end) in which case the story is about coming to terms with perhaps menopause, or a change of life.  To support that reading there is the parallel situation of the Ermine toque (a woman, like Miss Brill who is identified by the shabby fur she wears) whom Miss Brill watches with fascination as she first vainly attempts to interest a man, and then is rejected in one of the more chillingly brutal moments you might find in fiction.