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(I'm a bit concerned that I may have misinterpreted your question, since it feels rather trivial after my edits, and in light of the fact that $A^{\mathbb Z} \times B^{\mathbb Z}$ is naturally isomorphic to $(A \times B)^{\mathbb Z}$. But this does seem to be the standard definition of product cellular automata as used e.g. in doi.org/10.1016/j.ic.2008.03.012 and arxiv.org/abs/0902.1441, just to name two of the results I found based on a quick Google search.)
Hi, kiki, and welcome to MathOverflow. I've edited your question to (hopefully) standardize and clarify the notation a bit, and also to generalize it slightly to allow the original automata to have distinct alphabets $A$ and $B$. I hope I haven't introduced any errors while editing; if I have, please do correct them and feel free to improve or revert any changes I've made that you may disagree with. Thanks!
Technically, superstable patterns in the sense of the linked post (AIUI) don't have to be orphans, so their number on the lattice could increase over time. Assuming any existed, that is (which I personally suspect not to be the case for Conway's Game of Life, even if proving that seems quite difficult indeed). The number of superstable orphan patterns would indeed be an additive conserved quantity, though.
Note that the phrase "tile [a] square in the same way" needs to be interpreted carefully here. In particular, two tilings can have the exact same tile boundaries within the square and still not tile it "in the same way" in the necessary sense. (The infinite tile with one square missing from your other answer gives a nice counterexample — it can cover any finite square with no tile boundaries inside it!) Having the same boundaries for all tiles that overlap the square should be sufficient for two tilings to "tile the square in the same way", though.
… I've ran into something quite similar in applied math before: whereas a random family of $k$ permutations of an $n$-element set is quite useful as an ideal block cipher in cryptography, a random commutative family of $k$ permutations of an $n$-element set turns out, with probability tending to 1 as $k\to\infty$, to be extremely boring and useless.
This seems like an instance of a generic phenomenon where 1) we have a parametric family of objects we're interested in (e.g. groups with at most $n$ elements), and 2) it contains a subfamily (e.g. 2-groups) whose size grows faster with $n$ than the size of any other subfamily, such that for large $n$ almost all objects in the family belong to that particular subfamily, but 3) objects in the subfamily have properties that make their behavior very simple and boring, so that few if any of them are of any interest to anyone.
