Despite their poetic qualities, such uses of words as, "What if there was a universe with no logic?" seem to betray themselves by failing to reflect on what "if there was" is itself supposed to mean. Since normal logic includes an implicit theory of quantifiers and it is quantification that is assimilated to/over the concept of existence, asking if there was a universe without any logic could be to ask if there was a (as in at least one) universe without quantification at least in terms of the number 1.
So note also that prefix uni-: if this "other universe" cannot quantify over its contents, much less itself, as if it were "one universe," what sense would it make to talk about this universe being fully separate from or causally contiguous with ours? How can a universe be individuated without the concept of individuation?
Now Kant straddled the fence a lot, on this point. He accepted that the logical forms of judgment were not entirely localized in applicability, and would in some vague sense hold for objects of intellectual intuition. However, he does also float the abstract question of other forms of sensible (i.e. passive) intuition than space or time, and understanding/cognition without discursive conceptions. On top of all that, at one point he just claims that overly transcendent questions are basically meaningless, as if asking them is not to ask anything real at all. (A word of caution: those are paraphrases of an unsatisfactory (or not completely satsifactory) translation of Kant's. I say what I say and link to what I link to in good faith, but with an admission of tangential weakness in the presentation.)
Or consider Descartes' vexing/seeming claim that God could have created different eternal truths than He did. Maybe it's less that the law of non-contradiction could have been violated, and more that a world could have been created without enough propositional structure to maintain the purpose or significance of such a law; or no less that it could have been created with some other structure of greater significance. I think there's inescapable pushback, here, from the side of logical quantifiers, but it might not be the same problem (a logic without quantities of existence seems as if it would not itself exist (if, "Logic exists," is even much of a worthwhile statement), but a logic without non-contradiction is not as obviously pointless to imagine).