In my previous question, here: Can truths about the natural numbers vary across possible worlds?, I started off by saying that "The truths of logic are the same in all possible worlds". But is that really the case? Can there be possible worlds where there are different laws of logic? So, for instance, in the world we live in, classical logic is true, but in some other world, intuitionistic logic is true, and in yet another world, quantum logic is true. Also, have any philosophers written about this issue?
No. They absolutely can't. But they may or may not apply.
The reason I feel confident saying this is because any given logic is a set of axioms and some inference rules around how to apply them. If you change either the axioms or the inference rules, you have a new logic.
I think the question you are really asking might be: can there be a possible universe in which classical logic is not useful as a thinking tool? For example, is there a universe in which the law of identity is never a useful abstraction, because nothing is itself? I find that difficult to imagine, but my imagination is limited and I learned to think in a universe where identity is a useful abstraction.
Classical logic is not “true” in our world. We know this because classical logic is two state (true/false), but for for most questions we need at least 3 state logic (t/f/unknown) and for empiricism we need four state (t/f/u/nonsense).
So, logic in our world is pluralistic. There are an infinity of logics. And some approximate some aspects of our physical world fairly well, while others are needed for different aspects of it. So, we already are a Universe with the features you asked about.