Yes, this has been done in other settings. For example, Scott Balchin, Kyle Ormsby, Angélica M. Osorno, and Constanze Roitzheim wrote a paper, Model structures on finite total orders, that enumerates all model structures on a finite total order [n]. In fact, this paper is one in a series of recent papers on problems like this, usually with at least one of those four authors involved. For instance, the paper Composition closed premodel structures and the Kreweras lattice counts premodel structures and identifies model structures with certain tricolored trees. This area is starting to be called homotopical combinatorics, and at that link you can read a nice description by Kyle Ormsby about the connection between lattices and weak factorization systems.
I believe Inna Zakharevich had done earlier work counting model structures on posets, but I need to take some time to search for it. And, I think Andrew Salch told me about some work of his related to the nine model structures problem, perhaps extending it to vector spaces. I will look into my notes when I have time, and might edit this answer with more references if I find them.