how to formalize the class(?) of computational models and their equivalence
I recommend that you look at realizability theory. In realizability computational models are known as partial combinatory algebras (PCA). They cover a wide range of computational models. There is a 2-category of PCAs in which we can speak not only of equivlence, but also about general morphisms from one computational model to another.

Some references:


Jaap van Oosten's book on realizability.
John Longley's Ph.D. thesis.


John's Ph.D. is probably the best source to read about comparison of PCAs. An abridged version can be found in my Ph.D. thesis.

I should point out that realizability theory is just one way of organizing and structuring the world of computational models in a mathematically precise way. Nevertheless, it is a very sophisticated machinery which shows what can be done.