If another universe existed, would our logic still apply?

Logic is a cultural construct and we live among multiple cultures. There are logical extrapolations in Western culture that are not universally accepted outside the west (like “No one will defy us if we have the Hydrogen Bomb” or “Legitimate government derives from the consent of the governed”). You don’t need to go to an alternate universe to find people who disagree with these.

I wonder if the OP was actually asking whether it was inevitable that our universe condensed into the form with which we are familiar, or whether this is just one of many different possible configurations; for example configurations with an entirely different set of fundamental particles; with a different kind of spatial geometry etc…

Well with less than three spacial dimensions it becomes impossible to create very complex molecules, so life couldn’t reasonablt be expected to be able to achieve any inteligence in such a universe. For greater than three dimensions then complexity can be immence, but then energy would dissipate 1/r^3 or worse, so there are probably energy problems for such a universe.
More than one time like dimension would probably lead to such complexities that organised systems couldn’t be stable and if that were so life would be impossible for a universe with more than 1 time like dimension.

(site was Scientific American from about a year ago, where there was a big discussion about multiple universes and possibilities of this Universe being infinitely large)

It is not eh universe that obeys language, but me. If something did not follow logic in a broad sense, I would not be using language as I have learned it if I started describing it in otherwise normal terms. I mean nothing forbids me from calling summer leaves purple. Surely not the universe. But one might wonder if I had a grasp of what “purple” meant if I applied it so.