Like the other Educator said, the two female characters in both stories don’t necessarily suffer from the same mental illness but are rather pushed to do things they wouldn’t have normally because of their husbands’ actions.
The narrator in “The Yellow Wallpaper” likely experiences a form of postpartum depression with psychosis, by modern interpretations of her symptoms and the circumstances leading to her illness. Her husband, as a physician and man, insists that he knows what is best to treat what ails his wife, despite her repeated pleas for a different course of treatment and worsening condition. In this way, the narrator’s mental illness is only exacerbated by the patriarchal structure of marriage. Her agency is denied and she is at the mercy of her husband’s judgment, even in matters of health.
Similarly, Mrs. Wright is dominated by her husband. According to the conclusions of both Mrs. Hale and Peters, Mrs. Wright’s husband abused his wife and manipulated her psychologically; the death of her precious canary is cited as just one example of his abuse. The totalitarian authority of her husband and his repeated abuse of this authority drives Mrs. Wright to homicide as the only means of escape. While one might argue that any individual who kills another suffers from some type of mental illness, I don’t think it makes sense to say Mrs. Wright is mentally ill.
Rather, it could be said that both women suffer from pathological patriarchy.
This is a rather problematic question, as it is difficult to state with any certainty that any of the characters in "A Jury of Her Peers" suffer from mental illness in the same way as the narrator does in "The Yellow Wallpaper." In this short story, indeed, it is possible to argue that the problems experienced by the narrator are not due to any mental instability, but are actually a result of the restrictions that are placed on her by a patriarchal society, summed up in her husband, John, who insists on keeping her in the room with the yellow wallpaper. When her madness does strike, therefore, it is possible to see her instability as directly stemming from this sense of entrapment:
The pattern does move--and no wonder! The woman behind shakes it!
The way that the narrator sees a female figure trapped behind the wallpaper, and she images that this figure shakes against the bars or lines of this wallpaper, trying to get out, directly corresponds to her own condition as a woman trapped and intellectually stifled by her husband.
In "A Jury of Her Peers," there is a similarity in the way that women are obviously trapped and confined by marriage and having to work in their kitchens. Mrs Hale identifies the kind of lonely, harsh existence Minnie Wright would have led and she and Mrs Peters piece together the clues that they in their position as females are able to identify as directly incriminating Minnie Wright and showing that she killed her husband. Note, for example, what Mrs Hale says about the pressures of loneliness and how difficult Minnie Wright's llife must have been:
I might 'a' known she needed help! I tell you, it's queer, Mrs. Peters. We live close together, and we live far apart. We all go through the same things--it's all just a different kind of the same thing! If it weren't--why do you and I understand? Why do we know--what we know this minute?
She clearly admits that women suffer the same pressures of loneliness and the same struggles in life, and this is what she and Mrs Peters use to help understand how and why Minnie Wright murdered her husband. However, there is no indication that Minnie Wright suffered from mental illness. Rather, her act of violence was a completely natural response to having been entrapped and kept in a joyless world by her husband for so long. The link between Minnie Wright and the narrator in "The Yellow Wallpaper" is therefore not a shared mental illness, but their confined states thanks to their husbands, and how they suffer as a result.