Miss Brill's personality can only be analyzed with major bias considering that very little information is available about her current state of mind in the story of the same name, Katherine Mansfield's "Miss Brill".
From the information that we can gather from the story, we know that she does seem to feel a sense of connection and a form of community with the dynamics that she sees in the part.
In a way, we could say that Miss Brill is the epitome of atmosphere, rather than action and words, in the story. She is a stamp within the vastness of all that occurs in this Sunday afternoon without causing any effect or consequence for her presence.
She, a spinster teacher who has seemingly very little life experience to add to her wide sphere of things, is quite attached to her fur piece which she takes in her Sunday walks. It is quite possible that what we have here is a typical case of someone that is so used to being alone and following her own rules that she cannot fathom changing one thing about herself.
However, we know that Miss Brill is now aware that her life is as empty as it is lonely: When the teenagers at the park say that basically she is not welcome in the park, that she is outdated, and that she looks ridiculous. This is how Miss Brill realizes that she, like her little rogue, as she refers to her antiquated and old-fashioned rug, are alone in the world, are obsolete, and are only good when they are put away from the public view.
Certainly we can assume that Miss Brill is simply socially awkward and that such is the reason why she does not communicate with the external world. What could she possibly share with the others? What could she understand about other people's lives? She is safer by staying alone and by being her own biggest support system.