I think you've been misled by the standard description of combinatorial species. It's cleaner to think of (exponential) generating functions in one variable as being categorified by analytic functors, namely endofunctors of the form
$$X \mapsto \bigsqcup_{n \ge 0} F_n \otimes_{S_n} X^{\otimes n}$$
where $X$ is an object in a symmetric monoidal category $M$ with infinite coproducts and finite colimits, although these endofunctors won't behave as expected unless the monoidal product distributes over colimits. For example, pointwise coproduct corresponds to adding generating functions, pointwise tensor product corresponds to multiplying generating functions, and composition corresponds to composition.
A straightforward categorification of (again, exponential) generating functions in more than one variable is then analytic functors of more than one variable, e.g. in two variables these take the form
$$X, Y \mapsto \bigsqcup_{n, m \ge 0} F_{n, m} \otimes_{S_n \times S_m} X^{\otimes n} \otimes Y^{\otimes m}.$$
The relationship to species comes from describing convenient choices of "coefficients" $F_n, F_{n, m}$, which I didn't specify above. A common situation is that the coefficients live in a symmetric monoidal category $V$ such that $M$ is both enriched and tensored over $V$. Then $M$ naturally acquires an action by analytic functors with coefficients given by $V$-valued species, or equivalently presheaves $S^{op} \to V$, where $S$ denotes the groupoid of finite sets and bijections. For analytic functors in two commuting variables the coefficients come from presheaves $S^{op} \times S^{op} \to V$, etc. Here you should think of $S$ as the free symmetric monoidal category on an object, $S \times S$ as the free symmetric monoidal category on two objects, etc.
For noncommuting variables and ordinary generating functions, replace "symmetric monoidal" with "monoidal" everywhere above, except that I think extra conditions are required for the resulting functors to be closed under composition (e.g. $V$ should probably still be symmetric monoidal and it should act "centrally" on $M$). Coefficients are now given by $V$-valued presheaves on the free monoidal category on however many objects.
It's important to note that $S$ and $V$ have nothing to do with each other. The misleading thing about the classical case of combinatorial species is that $S$ and $V$ look very similar; the more general theory shows how they diverge. A very important special case is when $V$ is some variant of vector spaces; then $V$-valued species are Schur functors, which subsequently naturally act on any symmetric monoidal linear category with enough colimits.
