Suppose $(\mathcal{V},\otimes,1)$ is a symmetric monoidal category and $\mathbb{C}$ is a $\mathcal{V}$-category. I will deliberately avoid usual powerful assumptions (eg completeness/cocompleteness) on $\mathcal{V}$.
If $\mathcal{V}$ contains an object $0$ such that there is a family of morphisms $X\otimes 0 \to 0$ natural in $X$, it is easy to construct an enriched monoidal $\mathcal{V}$-category $\mathcal{M}(\mathbb{C})$ over $\mathbb{C}$ analogous to the construction of the free monoidal category on an ordinary category, as follows. Its objects are finite tuples of objects of $\mathbb{C}$, while the hom-objects are:
$$\mathcal{M}(\mathbb{C})((c_1,\dotsc,c_m),(c'_1,\dotsc,c'_n)) =\begin{cases}\mathbb{C}(c_1,c'_1) \otimes \cdots \otimes \mathbb{C}(c_m,c'_m) & \text{if }n=m; \\0 & \text{otherwise}\end{cases}$$
Composition is defined component-wise after applying symmetries in $\mathcal{V}$ to line things up. If $0$ is initial, we get a canonical factorization of any $\mathcal{V}$-functor from $\mathbb{C}$ to a monoidal $\mathcal{V}$-category $(\mathcal{E},\oplus,I)$ through a (strong) monoidal $\mathcal{V}$-functor $\mathcal{M}(\mathbb{C}) \to \mathcal{E}$. If I'm not mistaken, this factorization is unique and this really is the free monoidal $\mathcal{V}$-category over $\mathbb{C}$.
Question: Can free monoidal $\mathcal{V}$-categories exist under other circumstances?
It seems to me that the above hypothesis has at least some necessary components. Let $\mathbf{I}$ denote the unit $\mathcal{V}$-category, with one object $*$ and $\mathbf{I}(*,*) = 1$, the unit in $\mathcal{V}$. Suppose the free monoidal $\mathcal{V}$-category on $\mathbf{I}$ exists, say $(\mathcal{F},\oplus,I)$, and consider $Z := \mathcal{F}(I,*)$.
If given any object $H$ of $\mathcal{V}$ it is possible to construct a monoidal $\mathcal{V}$-category $(\mathcal{E},\oplus,I)$ containing an object $X$ such that $\mathcal{E}(I,X) = H$, we can deduce that $Z$ is at least weakly initial. For the time being I can only easily do this for objects in $\mathcal{V}$ which are idempotent with respect to $\otimes$.
Meanwhile, assuming free monoidal $\mathcal{V}$-categories exist on more general $\mathcal{V}$-categories, we should recover the existence of the claimed morphisms $H \otimes 0 \to 0$.
It feels like I might be missing counterexamples where more interesting things happen or a slick proof via a suitable Yoneda embedding. Any references where these ideas are discussed would be appreciated!
Update: I found a counterexample (see answers) which actually shows that the construction above need not even be a valid definition of a $\mathcal{V}$-category for general $\mathcal{V}$! I am now curious when free monoidal $\mathcal{V}$-categories exist under the additional hypothesis that every object of $\mathcal{V}$ can appear as a hom-object of some$\mathcal{V}$-category (which avoids that issue).
