So the question is, how do you get from Max Tegmark's mathematical universe where any mathematical structure is as valid as our universe, to something like the landscape of string theory where all the basic principles of physics are already in place? I agree that this is a huge reduction from something that requires unlimited information to describe, down to something that requires only a few thousand bits.
I first proposed an outline solution to that problem over twenty years ago before Tegmark had even made his ideas public. You can still find it on archive.org at this link
http://web.archive.org/web/19971011074729/http://www.weburbia.demon.co.uk/pg/tot.htm
I said more about it in the last essay and a little in this one.
My answer is that a principle of universality comes into play and a pre-geometric master-theory for physics emerges from the complexity of the system of all logical possibilities. Of course this is just a hypothesis and I cannot provide the mathematical details, but I think my arguments make the idea plausible. I would now identify this master-theory with something like M-theory but if M-theory is not right the idea may still work with whatever is right.
Note also that the few thousand bits only describes the low energy vacuum solution that determines the laws of particle physics in our cosmology. Within that solution many different histories are possible according to the laws of quantum physics, so the vastness of the original "multiverse" has not gone away. It is just being seen through the principle of universality that makes general principals of physics emerge.