What's in a Name

Zさん："Hi 雪ちゃん, sweetie! How are you?"
雪さん：「We need to talk.」
Zさん："What's up?"
雪さん：「This relationship isn't working out, I think we should break up.」
Zさん："I see"
雪さん：「... the thing is, it's not you, it's me. Understand?」
Zさん："So what you are saying is, your name isn't ゆき, but rather, みき?"
雪さん：「。。。」