AFAIK, every mathematical theory (by which I mean e.g. the theory of groups, topologies, or vector spaces), started out (historically speaking) by formulating a set of axioms that generalize a specific structure, or a specific set of structures.
For example, when people think of a “field” they AFAIK usually think of $\mathbb R$, or $\mathbb C$. A topology started out as a concept defined on $\mathbb R^n$ if I’m not mistaken.
But I’ve also seen cases where a certain structure has a natural topological structure, such as certain sets of propositions in first order logic. As far as I know, the people who formulated the axioms of a topology had no idea of this application. And the topological structure of a set of FOL statements is certainly conceptually vastly different from one on $\mathbb R^n$, certainly not two structures I would have expected to have such a deep commonality.
I would like to make a list of examples of mathematical structures that
Are interesting and well-behaved structures (e.g. not mere pathological counter examples)
satisfy the axioms of some mathematical theory in an interesting and nontrivial way,
But whose emergence is (conceptually/historically) very different from the structure of which those axioms were originally intended as a generalization.