It's not clear what "the structure we mean" means without a way of definitely identifying it in our mind. Categorical axioms just make this explicit.

Yes, that's true. I think it is good to have a logic with a semantics which allows you to clearly say what the "standard models" are. For those models categoricity should be achievable. But at the same time it is also clear that there will always be other models, for which categoricity doesn't hold.

Categoricity isn't a property of a model, it's a property of a set of axioms which only have "one model up to isomorphism", or one "structure" in the sense of structuralism.

As commonly defined, you are right, but if you have the ability to select among your models, then you can use that to sharpen your notion of categoricity by considering only a certain class of models instead of all models.

You can say for example, categorical with respect to a certain cardinality of the model, that is you are only considering models of that cardinality.

And just like that, if your semantic allows it, you can also say categorical with respect to standard models. But for that you of course need a clear definition of what "standard" means. First-order logic doesn't provide such a clear definition.