Why does Miss Brill treat her fur in a special way?
The text of the story does not flat out explain why Miss Brill treats her fur the way that she does. The reader must infer that information based on other character traits of Miss Brill.
She is an older woman. She is not married, nor does she have a roommate of any kind. I feel that she is a lonely woman. That is why she goes to the park to people watch. Miss Brill does not only watch the people, but she makes up stories and lives for those people based on her observations. I believe that she treats her fur like a living companion for precisely the same reason she goes to the park. She is lonely.
The fur, while inanimate, still gives her a sense of companionship and comfort. That's not a weird idea. Little kids have imaginary friends all of the time. Another example of this kind of friendship with an object can be found in the movie Cast Away with Tom Hanks. He's trapped on a deserted island. His best "friend" is Wilson . . . a volleyball.