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This is a very strong philosophical assumption.

Logic definitely exists as a method in human philosophy. You could argue that it's somehow "baked in" to the universe, whether by a creator or by some other cause, and that we only discovered it and didn't invent it. But the claim that it transcends our universe and would exist and be accessible from all other possible universes, even ones with different physics, different kinds of space and time (two temporal dimensions? who knows?) etc doesn't seem obvious in the slightest.

This is especially true when there are so many types of logic used in mathematics and philosophy to begin with: first-order or higher-order logic, constructive logic, Peano arithmetic, Zermelo-Fraenkel, ZF with Choice, etc.

Even once you have the axioms, they may support different models. Are just the axioms true universally, but models varying across universes? Is it possible that there are universes where all statements are true? Are there universes with Choice and others without it?

[mekoka]

Again, it might be best not to conflate logic and physics. Having different universes with different laws of physics is very conceivable to the mind. Logic is pure philosophy. It's an exercise in reasoning. No physical observation is required. The various kinds of logic you cite are different subsets that coexist without ambiguity under the same big umbrella. That is, it's still the same logic. If one law were to create ambiguity with another (which is the mechanism used in proofs), then at least one would be declared void, or some more work would be needed to explain the conditions that lead to the paradox. Logical truths have no order of precedence. Our minds simply use the ones it's more comfortable to reason about as a scaffold to uncover new ones. But they all exist together and are all equally true without ambiguity, including those we have not uncovered yet. If we posit that another universe has logical truths that differ from our own, they will not only be foreign to us, but they will also be totally inaccessible to our imagination in a way that makes any kind of sense (unlike alternate laws of physics), as being able to reason about them in our own universe then gives them validity and creates the aforementioned ambiguity. 1 + 1 must be equal to 2 when adding those two quantities. If you say it doesn't, you're either mistaken, or you're talking about a different subset of the same logic (bases, set theory, etc) where the same symbols are used to mean something different. That doesn't qualify as conflicting logic.