With theories that are categorical, it seems like you could say that the theory is about collections of objects (numbers, points, etc.) with a certain structure (the structure the standard models have). For instance, second-order $\mathsf{PA}$ seems to be about collections of elements ordered by a successor function such that there is a first element and no last, with addition and multiplication defined over the elements. Even with $k$-categorical theories, it seems like you could understand the theory as being about a given structure, once you fix the cardinality.
But what about theories that don't seem to have any intended interpretation? What should I think something like group theory is about? It isn't about systems of elements with a certain structure---since the structures studied are too varied. But then what is it about?
Is there a more general way of understanding what a mathematical theory is about that generalizes to non-categorical theories?
I'll try to give a partial answer to the question (as it was reformulated in the comments).
When moving away from ($\kappa$-)categorical theories, a natural next step in generality would be to consider complete first order theories $T$ . In this case, any two models of $T$ are elementary equivalent (not necessarily isomorphic).
The structural link between two models of $T$ is now given by the Keisler-Shelah Theorem: they must have isomorphic ultrapowers.