How to say something is "neither x nor y" in Japanese
e.g. "That is neither blue, nor pink. It's purple."
Would it be one of these?
「それは青やピンクじゃない。紫ですよ。」
「それは青でもピンクでもない。紫ですよ。」
Is there a different way to say it when objects or living things are the subjects? e.g. "That is neither a bed, nor a chair. It's a sofa." or "That is neither a dog, nor a rabbit. It's a cat"

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I think the latter one is better than the former one.
So, "neither x nor y" in Japanese = "x でも、y でもない。"

You can use as well even if objects or living things are the subjects.
たとえば、
今日、僕が一緒にいたのは、父でも母でもない。僕の兄だ。

Answer

Just because your last sentence was easier for me to think of a translation for, I am going to try it!
I am pretty sure it's going to be something like this:

それは犬と兎もじゃなくて猫です。
"That is neither a dog, nor a rabbit. It's a cat"

*I am still a Japanese learner so I tried what I could! I hope that was correct and helpful (^_^)

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