Wikipedia has article about structures which are used as the interpretation of first order logic https://en.wikipedia.org/wiki/Structure_(mathematical_logic). I feel that such structures are theories in some other formalism and that, generally speaking, each model of some theory of some logic is some other theory of some other logic and formalism. Isn't it so?
I just try to grasp if such uniform representation in the form "everything (theory and model) is some theory" is possible?
This can be of relevance for the uniform representation/encoding of the domains of discourse in the neural networks, i.e. I would like to propose that there should be no syntax/semantics difference (theory/model distinction) in the final effort to encode some domain in neural networks.
Information added. While, generally speaking, the distinction between theory and model is very well established, the fringe theories fails to find this distinction as represented by this paper https://www.cambridge.org/core/journals/mathematical-structures-in-computer-science/article/abs/locus-solum-from-the-rules-of-logic-to-the-logic-of-rules/6318E18EA633F9692D9CDBA9DE4438C9 which, with my deepest apologies involves highly unusual terms for the characterization of syntax-semantics confusion which can not be repeated here, where the persons with good manners are present.
You seem to be confusing terminology from mathematical logic (which is used to study the technical properties of formal deductive systems) with terminology from philosophy and AI, which deal with cognition.
The terms "theory" and "model" in mathematical logic have precise definitions: a theory is a set of sentences in a formal language, and a model is a structure that provides truth values for sentences in a specific way. These two kinds of objects have different shapes as sets, so it doesn't make sense to say that a model is a theory.
If you're arguing that theories and models (or syntax and semantics?) are in some sense the same thing, I think you're talking about something else and are doing philosophy, not math. In any case, it would help to define your terms and make your claim more precise.