The answer to "is it possible to unify them?" is almost certainly "yes" - I'm sure you can define some framework into which these examples (and probably many others) fit. But I doubt that such a definition would be useful or illuminating. Let me explain why.
First, attempts to make "universal" definitions often come in conflict with the endless creativity of mathematicians. If you suggest a general definition of "structured set" and "structure-preserving morphism", someone is likely to come along the next day and point out that the kind of object they're studying is a "structured set" that doesn't fit into your framework.
Second, even if you manage to come up with a sufficiently broad definition of "structured set", the answer to "Is there a way to define the morphism naturally when the structure is defined?" is going to be no, just because for some kinds of structures, there are multiple natural notions of structure-preserving morphisms. For example, should a structure preserving map between metric spaces be a metric map (weak contraction)? A uniformly continuous map? A Lipschitz map? etc. All of these are useful in different contexts. Similarly, in model theory it is useful to consider several different kinds of "structure-preserving" maps, including homomorphisms, embeddings, and elementary embeddings.
Finally, the goal of abstraction is to strip away irrelevant details, leaving only some simple properties, in order to illuminate patterns and techniques that apply to a wide variety of examples.
For example, in model theory, we study structures in the sense of sets equipped with operations and relations. This framework is certainly not universal: not every mathematical object can be naturally viewed as a structure in this sense. But nevertheless it is a natural setting to work in, which includes many examples. And this level of abstraction takes advantage of the fact that there are similarities between the way that algebraic structures and relational structures behave, and the point of view of definability in first-order logic provides tools to work with all structures of this kind.
Now there's a trade-off: If you widen your view further, you gain more examples, but you lose patterns and common techniques. The theories of your three examples (algebraic structures, topological spaces, and manifolds) are sufficiently different that the proper level of abstraction to encompass all of them is almost certainly category theory - by which I mean that the patterns and techniques common to all these areas don't depend on the fact that the objects are structured sets and that the morphisms are structure-preserving functions, but rather are purely categorical in nature.
To sum up: My view is that any "universal" definition of "class of structured sets and structure-preserving morphisms" would probably (a) fail to be universal in that it omits some interesting structures and some interesting morphisms, and (b) be quite complicated, thus not living up to the goal of abstraction: to sweep away the complicated details.
