As Emily is older than her sister, it is obvious she developed her acting abilities earlier. So, when they were young, she used to prank and bully Zooey into thinking absurd things.
Emily had Zooey convinced that "Neptunians had [taken] [her] real sister and replaced her with a facsimile who looked just like her."
Further elaborating on this, Emily said in an interview with Conan: "I would tell her… that I'd [hurt] her sister and taken over her body, and that I was going to kill her next." She would even make her cry and laugh afterward.
Eventually, Zooey got to a point where Emily had to do nothing but widen her eyes, and she would get scared.