MUH does sound exactly like what you're looking for in your OP. Regarding your remaining concern:
Tegmark they didn't write anything about substances and abstract/concrete objects, his arguments are mostly from a scientific perspective.
First of all MUH is entirely math based thus it describes anything from a fully scientific perspective, not even mostly. For the concrete objects and substances, including much advanced self-awareness, from MUH's wikipedia says:
Mathematical existence equals physical existence, and all structures that exist mathematically exist physically as well. Observers, including humans, are "self-aware substructures (SASs)". In any mathematical structure complex enough to contain such substructures, they "will subjectively perceive themselves as existing in a physically 'real' world".
Of course these are just some sketches and rough ideas from MUH to address your concern, not remotely realizable yet. There's been some patterns suggested from neuroscience studies such as re-entrant signaling or strange loop, etc. Here's some explanation about our perceived "causality" from strange loop theory:
Hofstadter thinks our minds appear to us to determine the world by way of "downward causality", which refers to a situation where a cause-and-effect relationship in a system gets flipped upside-down. Hofstadter says this happens in the proof of Gödel's incompleteness theorem:
Merely from knowing the formula's meaning, one can infer its truth or falsity without any effort to derive it in the old-fashioned way, which requires one to trudge methodically "upwards" from the axioms. This is not just peculiar; it is astonishing. Normally, one cannot merely look at what a mathematical conjecture says and simply appeal to the content of that statement on its own to deduce whether the statement is true or false.
Hofstadter claims a similar "flipping around of causality" appears to happen in minds possessing self-consciousness. The mind perceives itself as the cause of certain feelings ("I" am the source of my desires), while according to popular scientific models, feelings and desires are strictly caused by the interactions of neurons.
Regarding your other concern:
I've never seen modal realism being described in that way either.
Again from the same reference above, it suggests:
Jürgen Schmidhuber argues that "Although Tegmark suggests that '... all mathematical structures are a priori given equal statistical weight,' there is no way of assigning equal non-vanishing probability to all (infinitely many) mathematical structures." Schmidhuber puts forward a more restricted ensemble which admits only universe representations describable by constructive mathematics, that is, computer programs; e.g., the Global Digital Mathematics Library and Digital Library of Mathematical Functions, linked open data representations of formalized fundamental theorems intended to serve as building blocks for additional mathematical results.
In response, Tegmark notes that a constructive mathematics formalized measure of free parameter variations of physical dimensions, constants, and laws over all universes has not yet been constructed for the string theory landscape either, so this should not be regarded as a "show-stopper".
Don Page has argued that "At the ultimate level, there can be only one world and, if mathematical structures are broad enough to include all possible worlds or at least our own, there must be one unique mathematical structure that describes ultimate reality. So I think it is logical nonsense to talk of Level 4 in the sense of the co-existence of all mathematical structures." This means there can only be one mathematical corpus. Tegmark responds that "this is less inconsistent with Level IV than it may sound, since many mathematical structures decompose into unrelated substructures, and separate ones can be unified."