I came in far too late to join the discussion below, and it's not easy to branch off into a different mode of reasoning.
What physical attributes would both Univeres's share?
Or prehaps what would be the biggest difference.

An explanation of the fundamentals could help give one a deeper insight into whether there is a Multi Universe state at all.

In multiverse theory there would not necessarily be any relationship between the different universes.

One could potentially argue that there might be some universal constant that would span them but the argument could just as easily be made the other way too.

DA Morgan is right. The difference is that a multiverse theory could (in principle) explain things that are (even in principle) unexplainable in single universe theories.

A multiverse theory can be defined by a probability distribution over some set of allowed universes. Assuming that this set is large enough you will always find yourself alive in conditions that are consistent with your existence.

This seems trivial, but this ''Anthropic Principle'' is sometimes used in astrophysics. However, in single universe theories it has no deeper explanation, while in generic multiverse theories it is a consequence of the theory itself.

The Anthropic Principle is actually superseeded by a more powerful principle in multiverse theories. You would expect to find yourself in the most likely circumstances. The predictive power of multiverse theories is thus greater than that of single universe theories.

It is purely Boolean that we exist in an environment in which we can exist.

Arguments to the contrary are pure nonsense.

That said those that engage in soft-heading pursuits such as pychology and theology often use this to build a house of a foundation of quicksand for those with less agile minds.

A Physical relationship? No, for its an invisible multiverse (we do not see or feel it)
But its certainly made out of the same materials
as the universe we presently live in. Its matter may well be infinitly more compact, or infinitly further apart. But whether its folded or open, It must be made of the same materials as our present Universe. Were it not, the whole multi-universe concept would fail. Our life upon a multiuniverse included. I am looking/asking for logical answers. Rather than the theoretical idea of an infinite number of universes.
Time might be compacted or stretched, but I believe certain laws, or theorems should still hold. Many answers would still give a Constant, prehaps a different constant than ours, but a constant nevertheless. A circle, would still look and measure, as a circle, even within a Swartzchilde radius. The conservation of matter, also, would still hold, a zero still a zero.
The 3rd, 4th, 5th, ....11th and higher dimensions of the universe could be under stress. They might be running down? Are we a product of the (final) 3rd or 4th dimension? Creating or destroying a higher dimension will surely follow fixed laws. Unknown laws, that we can only guess at.
Matter is matter. Stretch it, fold it, compress, push or pull it, matter it remains.