Is this a helpful analogy?
Any categorical theory (whose models are all isomorphic) describes a structure uniquely up to isomorphism. In this case there is no need to start with a set-model and forget about it after it has done its work.
Question (maybe dumb): Are there infinite categories that are the unique-up-to-isomorphism model of a categorical theory?
On the other hand: Only non-categorical theories give rise to full-blown concrete categories like that of groups with homomorphisms, topological spaces with continuous maps and so on.