An axiomatic theory of computational models?
The question "what is a computational model" is quite open ended and is difficult to answer definitely. It's a bit like asking "what is geometry". But people have given some answers.

As far as I can understand you, you are looking for a general theory of computation (so, not a model that "contains other, for what's that supposed to mean, nor an "axiomatization" of a specific thing). One such theory is offered by partial combinatory algebras (PCA), which are very simple structures that subsume many models of computation, including: Turing machines (with and without oracles), $\lambda$-calculus (with many of its extensions), enumeration operators, and even topological models of computation. There are further variants of PCAs: ordered PCAs, typed PCAs, etc.

One should never just "collect models of computation" but also ask how to relate them, so we should think about transformations between PCAs. John Longley gave a very good account of these, namely the applicative morphisms. Speaking a bit vaguely, an applicative morphism is an interpretr of one PCA implemented in another PCA.

If you are interested in this topic you can consul Jaap van Oosten's book on realizability.