Here is a proposal that avoids requiring a notion of free monoid. Let $M$ be a monoid in some monoidal category $C$ and let $s : S \to M$ be a morphism (there is really no reason to restrict our attention to subobjects / monomorphisms).

**Definition #1:** $S$ *weakly generates* $M$ if, for any parallel pair of morphisms of monoids $f, g : M \to N$, for $N$ a monoid, $f \circ s = g \circ s$ implies $f = g$.

That is, $s$ is right cancellable with respect to morphisms into other monoids. If you have a notion of free monoid, by which I mean a left adjoint $F$ to the forgetful functor from monoids in $C$ to $C$, then this is equivalent to the induced map $F(S) \to M$ being an epimorphism of monoids.

This definition has the substantial drawback that it already fails to reproduce the usual notion of generation for rings! The problem is that an epimorphism of rings need not be surjective. Here is a second proposal which at least reproduces the usual notion of generation for rings (more precisely, it reproduces the notion of an additive subgroup of a ring generating it under multiplication), but now I need to assume there is a left adjoint to the forgetful functor, and probably I should also assume that coequalizers exist or else the proposal will probably behave strangely.

**Definition #2:** $S$ *generates* $M$ if the induced morphism $F(S) \to M$ is a regular epimorphism; that is, if there is some other object $R$ and a pair of morphisms $f, g : R \to F(S)$ such that the induced morphism $F(S) \to M$ is their coequalizer.

This definition has the benefit that the object $R$ can be regarded as specifying relations that the generators satisfy. It also reproduces the usual notion of generation for rings. (In the case of rings it doesn't matter whether you require the morphism to be a regular epimorphism of rings or just a regular epimorphism of abelian groups; I am not sure whether the distinction matters in general.)

A more canonical definition that doesn't rely on assuming that some auxiliary object exists is given by the notion of effective epimorphism. Fortunately the distinction vanishes in a category with pullbacks and so is not too important in practice.

Both definitions are manifestly categorical and so manifestly invariant under monoidal equivalence of monoidal categories (which I presume is what you meant to ask).

Finally, it is probably worth remarking that if $C$ has countable coproducts and the monoidal structure distributes over countable coproducts in both variables then the free monoid functor $F$ can be constructed very explicitly as

$$\displaystyle S \mapsto F(S) = \bigsqcup_{n \ge 0} S^{\otimes n}.$$

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