The real question here is what properties the language of category theory captures.
A statement in the (finitary) language of category theory is one formed from propositions of the form $f = g$ where $f, g : X \to Y$ using the propositional operations of $\lor, \land, \top, \bot, \neg, \implies$, together with quantifiers either over all objects (eg $\forall X$, $\exists X$ where $X$ is an object variable) or over all morphisms between two given objects (eg $\forall f : X \to Y$ or $\exists f : X \to Y$ where $X, Y$ are objects).
Two things can be shown about statements in the language of category theory. Suppose given a statement $\phi(X_1, ..., X_n, f_1, ..., f_m)$ in the language of category theory with object variables $X_1, ..., X_n$ and free function variables $f_i : X_{d_i} \to X_{c_i}$ for $1 \leq i \leq m$.
Now consider a category and two different variable assignments - one assignment $X_1 \mapsto W_1, ..., X_n \mapsto W_n$ where each $W_i$ is an object, and $f_1 \mapsto g_1, ..., f_m \mapsto g_m$ where $g_i : W_{d_i} \to W_{c_i}$ for all $i$, and another assignment $X_1 \mapsto Y_1, ..., X_n \mapsto Y_n$ and $f_1 \mapsto h_1, ..., f_m \mapsto h_m$, where $Y_i$ is an object for all $i$ and $h_j : W_{d_j} \to W_{c_j}$ for all $j$. Suppose there are isomorphisms $k_1 : W_1 \to Y_1, ..., k_n : W_n \to Y_n$ such that for all $i$, $h_i = k_{c_i} \circ f_i \circ k_{d_i}^{-1}$. Then $\phi(W_1, ..., W_n, g_1, ..., g_m) \iff \phi(Y_1, ..., Y_n, h_1, ..., h_m)$. This is known as "isomorphism invariance of truth".
Now consider a functor $F : C \to D$ which is fully faithful and essentially surjective, and a variable assignment in $C$ of the form $X_i \mapsto W_i$, $f_j \mapsto g_j$ as above. We assume that $\phi$ has no free variables other than the $X_i$ and $f_i$. Then $\phi(W_1, ..., W_n, g_1, ..., g_m)$ iff $\phi(F(W_1), ..., F(W_n), F(g_1), ..., F(g_m))$. This is known as "equivalence invariance of truth".
Both of the above can be proved by induction on formulas.
The two above statements can be generalised to the infinitary language of category theory, which allows quantification over external sets, though some care must be taken in settings without the axiom of choice.
So the language of category can only discuss properties which are invariant up to isomorphism and up to equivalence of categories. No finer-grained discrimination is possible.
For example, consider the category of sets. It follows from isomorphism invariance that the language of category cannot distinguish two sets which have the same cardinality. The language of category cannot distinguish a monic $f : S \to T$ from a subset $S \subseteq T$.
For the category of topological spaces with morphisms the continuous maps quotiented by homotopy, the language of category theory cannot distinguish between homotopy equivalent spaces.
So the key when using category theory is to find the right amount of data for morphisms to carry based on the underlying subject one is studying and the particular context.
Best Answer
It is by definition that the maps in the category of vector spaces are linear maps. The full description of the data defining the category of vector spaces and linear maps is, after fixing a base field (such as the real numbers $\mathbb{R}$):
Now you need to check that this data is indeed a category: that composition is associative, that composition lands in the correct set (for the above example, we need $f \circ g$ to be a linear map $V \to W$), and that an identity morphism $\operatorname{id}_V \in \operatorname{Hom}(V, V)$ exists for each object $V$.
Here are some other examples of random categories we could define:
There are many things that are categories, some useful, and some not. But the defining data are objects, morphisms, and composition. Then you need to check that the conditions hold. Here are some non-examples of categories: