The first answer to your question is "yes". In a topos, say, morphisms are in bijective correspondence with functional relations. More precisely, given objects $A$ and $B$, there is an isomorphism between the exponential $B^A$ and the object $FunRel(A,B)$ of functional relations between $A$ and $B$. In the internal language of the topos $FunRel(A,B)$ is described as (where $\exists!$ means "exists exactly one") $$FunRel(A,B) = \lbrace F \in \mathcal{P}(A \times B) \mid \forall x \in A \exists! y \in B . \; \langle x, y \rangle \in F \rbrace.$$ You will of course recognize this as the usual set-theoretic definition of a function as a collection of ordered pairs. The crucial bit of logic that is needed for the bijection is the principle of unique choice, which says that for every $R \subseteq A \times B$, $$(\forall x \in A \exists! y \in B . \; \langle x, y \rangle \in R) \implies \exists f \in B^A \forall x \in A . \langle x, f(x) \rangle \in R.$$
This brings me to the second answer to your question, which is "no". Typically, the definitions of functions which do not rely on set theory and first-order logic are more widely applicable that the usual set-theoretic one. Two examples come to mind:
- the notion of morphism can be interpreted in any category, as you already mentioned,
- the definition of function as an element of a function type $\tau \to \sigma$ in a (simply typed) $\lambda$-calculus allows us to interpret functions as programs, see for example primitive recursive functionals. These do not in general satisfy the law of extensionality $(\forall x : \tau .\; f(x) = g(x)) \implies f = g$.
Students of mathematics are sometimes told that in the old days people were confused about the notion of function, that they equated functions with symbolic expressions (like students still do), or with Taylor expansions (not the same Taylor). The students walk away with the impression that those silly 17th, 18th and 19th century mathematicians lived in great confusion, thinking that all functions can be computed and they all have series expansions, and that we all have to thank 20th century set theory for setting us straight. But that's a rather narrow view of the world. There are very good reasons why in certain contexts we should think of functions as something other than bags of dust. Sometimes function are symbolic expressions (called "programs" nowadays), and sometimes they are Taylor expansions (in synthetic differential geometry). To paraphrase a Taylor, in such situations it is better to axiomatize the algebra of functions directly than to glue them together from dust found in some bags.
