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...
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