First we should remember that morphisms in a category do not necessarily have any resemblance with a structure-preserving map. Think of the homotopy category for example, or the category of paths in a directed graph.
The notion of "structure-preserving" pops up in several places, and in each case it can have a slightly different precise meaning. Many examples can be grouped together, though, by using more general constructions of categories.
For example, the usual algebraic structures are actually algebras of a monad. If $T$ is a monad on a category $\mathcal{C}$ and $(x,a)$, $(y,b)$ are two $T$-algebras, then a morphism $x \to y$ can be called structure-preserving, or a morphism of $T$-algebras, if the diagram
$$\begin{array}{ccc} T(x) & \rightarrow & T(y) \\ \downarrow && \downarrow \\ x & \rightarrow & y \end{array}$$
commutes. Basically, the morphism $T(x) \to x$ is the structure on $x$, likewise for $y$, and the diagram tells us that $x \to y$ preserves the structures. We get a category of $T$-algebras. Starting with $\mathcal{C}=\mathbf{Set}$, we get the categories of monoids, groups, rings, lattices, etc. We may also start with $\mathcal{C} = \mathbf{Top}$ and get the categories of topological groups, topological rings, etc.
The same discussion applies to comonads and their coalgebras.
A functor between two categories can be seen as a pair of maps between objects and morphisms which are structure-preserving. Also, a natural transformation between two functors can be seen as a family of maps which is structure-preserving.
The category of topological spaces $\mathbf{Top}$ has also structure-preserving morphisms in the following sense: a map $f : X \to Y$ between topological spaces is continuous if and only if for all nets $x : P \to X$ with a limit $u$ the net $f_*(x) : P \to X \to Y$ has limit $f(u)$. In short, $f$ commutes with the operation of taking limits. And a topology can actually be defined in terms of net convergence. This even shows that $\mathbf{Top}$ is the category of models of a limit sketch, see arXiv:2106.11115. Morphisms between models of a limit sketch are just natural transformations, so are structure-preserving maps as mentioned.
Relational structures also have obvious definitions of structure-preserving morphisms. This gives the categories of partial orders, linear orders, etc.
And finally, if a category already has structure-preserving maps as morphisms, this is of course true for every subcategory. This yields the category of fields, for example, from the algebraic category of commutative rings.
So in essence, no, there is no general definition of the term "structure-preserving". It depends on what type of structure you are looking at. Also, there might be different answers (what is a structure-preserving map between normed vector spaces?).
But if you really want to go abstract nonsense, the Yoneda Lemma is similar to what you want: the structure on an object $x$ of an arbitrary category is the collection of all morphisms $t \to x$ and how they relate to each other. To be precise, it is the hom-functor $\hom(-,x)$. You could also take the hom-functor $\hom(x,-)$ if you like. The Yoneda Lemma tells us that a morphism $x \to y$ is the same as a natural transformation $\hom(-,x) \to \hom(-,y)$. The latter is a bunch of maps $\hom(t,x) \to \hom(t,y)$, thus assigning to each structure on $x$ a structure on $y$ (of the same type). In this sense, you might call morphisms structure-preserving (but I would not recommend it).
