Not knowing the protagonist's name represents the anonymity of a woman in a marriage with a repressive and dominating man (A).
The Yellow Wallpaper was written as a critique of the treatment of young wives in typical 19th-century marriages, often patronized by husbands and male doctors as "hysterical." In the story, the main character's husband isolates her in a summer house for what he believes to be her best interest, while preventing her from working.
This lack of social life outside the home and the impossibility to write is what the author, Charlotte Perkins Gilman, denouces as ways to silence women, limit their means of expressing their individuality, and keep their emotions under control. This is possibly why her character is stripped of an identity by having no name.