This is idealism getting lost in the sauce to the highest degree. If you ever find yourself thinking "reality is just math", you must remember: the map is not the territory.
In the paper he advocates going farther and saying, there's no independent "space" just the vector and space(time) emerges purely from the evolution of that vector.
If your territory is a map, then the map might be the territory.
Magic. Reversing the relationship between reality and our interpretation of it literally means bending reality with ideas. In this case magic is matrix multiplication.
But if you get predictions from the theory that turn out to be true in the end (like entanglement, Bell's inequality, etc) then it gives you a lot of confidence in your theory.
I don't know how it'll turn out. At this point people are throwing stuff against the wall to see what sticks to get physics out of the local minima it seems to have found itself in.
I don't think Carroll or anyone else is naively wandering into this endeavor from a philosophical standpoint though.
Like if on a low level it turns out that everything is, I don't know, cellular-automata-based, that would be a more useful frame than wandering vectors in a Hilbert space. But if we get to the end of physics and found that, in fact, you could rederive everything from that one explanation, it wouldn't be unfair to say that the universe is that vector, and perhaps some functions.
I guess really I'm wondering if it matters more how the universe is 'computed' or what it's 'computed on'. In a non-simulated universe, of which it seems there must exist at least one, there would at some point just have to be laws without causes. If those laws corresponded to some simple bit of math, it wouldn't be wrong to merge map and territory.
edit: That's not to say that I actually think the paper is correct. I'm definitely not far enough along in my education to be 100% sure, but there are enough suspicious features and a high enough bar that I'm doubtful. I just meant to mount a general defense of 'what if it's all math' type explanations.
Would it be metaphysical? Not really, because it does not speak about anything but the computation which, by construction, is the reality of such universe. It claims no knowledge of the computer implementation details which are mot reflected in the computation, like its power consumption or the color of the chassis, or even the fact that a chassis might exist.
I think that sticking to intuitive and customary notions of reality when experimental data contradict them is not scientific. Science is all about building a better model that fits more and more experimental data points as tightly as possible. If such model introduces notions that the mundane experience entirely lacks, while still describing these mundane phenomena perfectly, it's time to admit that the reality is indeed "weird", and it's your mundane intuitions which are wrong (or, rather, apply narrowly).
Also, the end result of science is the math that fits the experiments, that is, the model of reality, the language of it is math. You can ask whether there's something behind that math, like you might ask whether there's something behind the computations in a virtual reality universe. I assume we can't see past this border with scientific means, only with metaphysical.
What makes you think your perception of reality is not the map in this dichotomy?