What level of social class does Miss Brill project?
Judging from the story, one would argue that Miss Brill projects someone with education but without the financial means to use her knowledge to conduct more sophisticated activities. Hers is not abject, but genteel poverty; she certainly does not have much, but does what she can with what she has the most of, all for herself: time.
We know that Miss Brill is an English expatriate living in France, where she works as an English teacher. She does not hold expensive lodgings, and lives in a room, which may mean that she is a boarder.
But to-day she passed the baker's by, climbed the stairs, went into the little dark room - her room like a cupboard - and sat down on the red eiderdown.
Boarding, at least back in the time period where the story takes place (early 1920's) was the least expensive lodging option available for a single person. Think about the novel The House of Mirth and the horror that its protagonist, Lily Barth, felt at the prospect of becoming so poor that she would end up boarding up somewhere.
The fur she is so fond of, which is in questionable conditions, has not been yet fixed.
...the nose, which was of some black composition, wasn't at all firm. It must have had a knock, somehow. Never mind - a little dab of black sealing-wax when the time came - when it was absolutely necessary..
The need to wait for it to be fixed may have a sentimental undertone as to not alter or "hurt" this inanimate object to which she feels a connection. However, it could also be interpreted as a lack of the needed resources, the black sealing-wax and the money to acquire it, that accounts for the fur's condition.
On and all, Miss Brill and her fur have a lot in common. They were once in a "better shape". As a rule of life, time took a huge dig at them, and the lack of resources answer to her overall lack of detail in things such as fixing her old fur, or even getting a new one. All of this does not take away any of her dignity, but reveals an even sadder dimension of her character of which she has no control.