Here's a repair of the definition suggested in the OP. Namely, we need the additional assumption that $\text{Hom}(1, -) : C \to \text{Set}$ is faithful (so concretizes $C$). Then with the condition described in the OP, we get a map of sets $^{-1} : \text{Hom}(1, G) \to \text{Hom}(1, G)$, and now it really makes sense to ask whether this lifts to a morphism in $C$; faithfulness ensures that if such a morphism exists, it is unique, and therefore really a property of $G$.

So it may happen that $^{-1}$ exists but is not a morphism in $C$ (and that this is the sense in which $G$ is a group object), but also that $\text{Hom}(1, -)$ factors through some other category $D$ in which it is a morphism. For example, if $C$ is the category of Poisson manifolds, then apparently $D$ is the category of smooth manifolds. This provides some motivation for the definition in Ryan Reich's answer, and actually it suggests an even weaker definition:

Let $C, D$ be categories with finite products and $S : C \to D$ a faithful, product-preserving functor. A (let's say) $D$-virtual group object is a monoid object $G \in C$ together with a morphism $i : S(G) \to S(G)$ such that $S(G)$ (with the induced monoidal structure) is a group object in $D$ with inverse $i$.

I don't know whether this implies that there needs to be a good notion of heteromorphism associated to $i$ (unless by "heteromorphism" you mean nothing more and nothing less than a morphism in $D$).

Here's a repair of the definition suggested in the OP. Namely, we need the additional assumption that $\text{Hom}(1, -) : C \to \text{Set}$ is faithful (so concretizes $C$). Then with the condition described in the OP, we get a map of sets $^{-1} : \text{Hom}(1, G) \to \text{Hom}(1, G)$, and now it really makes sense to ask whether this lifts to a morphism in $C$; faithfulness ensures that if such a morphism exists, it is unique, and therefore really a property of $G$.

So it may happen that $^{-1}$ exists but is not a morphism in $C$ (and that this is the sense in which $G$ is a group object), but also that $\text{Hom}(1, -)$ factors through some other category $D$ in which it is a morphism. For example, if $C$ is the category of Poisson manifolds, then apparently $D$ is the category of smooth manifolds. This provides some motivation for the definition in Ryan Reich's answer, and actually it suggests an even weaker definition (edit: which I see Ryan has already suggested!):

Let $C, D$ be categories with finite products and $S : C \to D$ a faithful, product-preserving functor. A (let's say) $D$-virtual group object is a monoid object $G \in C$ together with a morphism $i : S(G) \to S(G)$ such that $S(G)$ (with the induced monoidal structure) is a group object in $D$ with inverse $i$.

I don't know whether this implies that there needs to be a good notion of heteromorphism associated to $i$ (unless by "heteromorphism" you mean nothing more and nothing less than a morphism in $D$).

Here's a repair of the definition suggested in the OP. Namely, we need the additional assumption that $\text{Hom}(1, -) : C \to \text{Set}$ is faithful (so concretizes $C$). Then with the condition described in the OP, we get a map of sets $^{-1} : \text{Hom}(1, G) \to \text{Hom}(1, G)$, and now it really makes sense to ask whether this lifts to a morphism in $C$; faithfulness ensures that if such a morphism exists, it is unique, and therefore really a property of $G$.

So it may happen that $^{-1}$ exists but is not a morphism in $C$ (and that this is the sense in which $G$ is a group object), but also that $\text{Hom}(1, -)$ factors through some other category $D$ in which it is a morphism. For example, if $C$ is the category of Poisson manifolds, then apparently $D$ is the category of smooth manifolds. This provides some motivation for the definition in Ryan Reich's answer.

Here's a repair of the definition suggested in the OP. Namely, we need the additional assumption that $\text{Hom}(1, -) : C \to \text{Set}$ is faithful (so concretizes $C$). Then with the condition described in the OP, we get a map of sets $^{-1} : \text{Hom}(1, G) \to \text{Hom}(1, G)$, and now it really makes sense to ask whether this lifts to a morphism in $C$; faithfulness ensures that if such a morphism exists, it is unique, and therefore really a property of $G$.

So it may happen that $^{-1}$ exists but is not a morphism in $C$ (and that this is the sense in which $G$ is a group object), but also that $\text{Hom}(1, -)$ factors through some other category $D$ in which it is a morphism. For example, if $C$ is the category of Poisson manifolds, then apparently $D$ is the category of smooth manifolds. This provides some motivation for the definition in Ryan Reich's answer, and actually it suggests an even weaker definition:

Let $C, D$ be categories with finite products and $S : C \to D$ a faithful, product-preserving functor. A (let's say) $D$-virtual group object is a monoid object $G \in C$ together with a morphism $i : S(G) \to S(G)$ such that $S(G)$ (with the induced monoidal structure) is a group object in $D$ with inverse $i$.

I don't know whether this implies that there needs to be a good notion of heteromorphism associated to $i$ (unless by "heteromorphism" you mean nothing more and nothing less than a morphism in $D$).