By the time the narrator's husband, John, returns to the house where his wife is being "treated" with the rest cure for her "hysteria"—a sort of catch-all term to describe anxiety or depressive disorders in women at the time—she is no longer herself. She's essentially had a complete mental breakdown where she now believes that she is the woman that she believes she has freed from the wallpaper. Unable to free herself and held practically a prisoner by her husband (who had good intentions but terrible methods), perhaps it is easier to believe that she is a free woman, even if it isn't the objective reality. She says to John,
"I've got out at last [...] in spite of you and Jane[.] And I've pulled off most of the paper, so you can't put me back!"
We've never heard a reference to a "Jane" before—we've met John, the narrator's husband, and Jennie, her sister or sister-in-law—and so it makes sense to assume that Jane is actually the narrator's name....
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