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][1], 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][2] counts premodel structures and identifies model structures with certain tricolored trees. This area is starting to be called [homotopical combinatorics][3], 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.

EDIT: I found another paper in homotopical combinatorics. In [Self-duality of the lattice of transfer systems via weak factorization systems][4] the authors prove a correspondence between $G$-transfer systems (where $G$ is a finite group) and weak factorization systems on the poset category of subgroups of $G$. I also wanted to point out that a lot of this work has been done by undergraduates at Reed College under the supervision of Ormsby and Osorno, and I find that super impressive.

Also, I found the paper of Inna Zakharevich (and Jean-Marie Droz it turns out): [Extending to a model structure is not a first-order property][5]. As the abstract says, they "characterize all model structures where $\mathcal{C}$ is a partial order." That paper builds upon an earlier one by the same authors, [Model categories with simple homotopy categories][6]. Related work of Droz ([Quillen model structures on the category of graphs][7]) proves that there are uncountably many model structures on the particular category of graphs she works with.

Andrew Salch made me aware of these papers, and also told me that there are five model structures on the category of vector spaces (a fact that was also known to Tom Goodwillie, as the comments below this answer make clear). In the notes of my conversation with Andrew, I find several more interesting statements of the type the OP is looking for. However, looking through Andrew's webpage and arXiv, it seems he hasn't published these statements yet. Therefore, I encourage the OP, and anyone else interested in these kinds of questions, to email Andrew.

EDIT 2: Just today, Andrew Blumberg, Mike Hill, Kyle Ormsby, Angélica Osorno, and Constanze Roitzheim [published a paper in the AMS Math Monthly][8], summarizing the new field of homotopical combinatorics. In the section Model Structures on Posets, they discuss how lattices parameterize weak factorization systems and allow for the counting of model structures. That article will surely be the most complete literature survey available at present.


  [1]: https://arxiv.org/abs/2109.07803
  [2]: https://arxiv.org/abs/2209.03454
  [3]: https://kyleormsby.github.io/posts/2021/09/homotopical-combinatorics/
  [4]: https://arxiv.org/pdf/2102.04415.pdf
  [5]: https://arxiv.org/abs/1410.6127
  [6]: http://www.tac.mta.ca/tac/volumes/30/2/30-02.pdf
  [7]: https://arxiv.org/abs/1209.2699
  [8]: https://www.ams.org/journals/notices/202402/noti2882/noti2882.html?adat=February%202024&trk=2882&galt=none&cat=interest&pdfissue=202402&pdffile=rnoti-p260.pdf