Is there a sensible way to enrich over SymMonCat such that id_X is not the monoidal unit? SymMonCat is the cartesian 2-category of symmetric monoidal categories, braided monoidal functors, and monoidal natural transformations.  The terminal symmetric monoidal category 1 has one object $I$ and $I \otimes I = I$.
A category enriched over a monoidal category $V$ assigns to each pair of objects $X, Y$ an object hom$(X,Y)$ in $V$ and to each object $X$ a morphism $id_X:I \to \mbox{hom}(X,X)$ in $V$.
When $V = $ SymMonCat, the morphism $id_X:1 \to \mbox{hom}(X,X)$ is a braided monoidal functor; since monoidal functors preserve the monoidal unit and tensor product, it must map the unit $I$ in 1 to the unit $I$ in hom$(X,X)$.
Is there a different way of enriching over SymMonCat such that $id_X$ does not pick out the monoidal unit (other than considering it a subcategory of Cat)? 
 A: [Ignore this first part, I'm just leaving it for the context to the comments below.] 
It is hard for me to understand why you would want to enrich in symmetric monoidal categories, have an identity, and also  want this identity to not be the unit of the symmetric monoidal category. 
That said, you can always do away with units altogether and consider "enriched categories without identities". Is this what you are after?

After Mike's example I am now on board. What you probably want to do is enrich over the symmetric monoidal 2-category of symmetric monoidal categories where the monoidal structure is the "tensor product of symmetric monoidal categories". What is this you ask? 
The functor category between two symmetric monoidal categories $Fun^\otimes(B,C)$ is naturally equipped with a symmetric monoidal structure (using pointwise multiplication). The tensor product of symmetric monoidal categories is $(-) \otimes B$ is the (weak) left adjoint to the functor $Fun(B, -)$. Thus $A \otimes B$ is a symmetric monoidal category such that symmetirc monoidal functors from it to C are the same as "bilinear" functors $A \times B \to C$. Now the monoidal unit for this tensor product is the free symmetric monoidal category on one object $\mathbb{F}$ (which is the category of finite sets and permutations). 
In this way, if you enrich in (SymCat, $\otimes$) you get a unit being a functor $ \mathbb{F} \to Hom(a,a)$, which is  equivalent to just some element, not necessarily the unit object of $Hom(a,a)$. 
The prototypical example is the 2-category of symmetric monoidal categories itself. 
A: I suppose that the monoidal structure for $ SymMonCat$ you  mean is the cartesian one, and  as  braided functor you mean a pseudo-monidal symmetrical (funtors that commutes with the canonical isomorphism of symmetry, and the coherence morphism data are isomorphisms). Then $\mathcal{C}(X, X)\in SymMonCat$ has a internal symmetric monoidal structure and another monoidal structure given by monoidal moltiplication $\mathcal{C}(X, X)\times\mathcal{C}(X, X)\to \mathcal{C}(X, X)$ with the realtive axioms related to a $SymMonCat$-enriched category.
Then $\mathcal{C}(X, X)$ is a bimomoidal category, and the two “monidal Identity" for the two monoidal structures are $I_X$ (for  the internal monoidal structure) and the (essential) image of the morphism you  called $id_X$, and a very reduced example of this (de-categorification)  is a $rig$, a algebraic structure by two monoidal law, one abelian. In general the two units are different (trivial example $(\mathbb{N}, +, 0 ; \times , 1)$.
A: A monoidal functor preserves the unit and tensor only 'laxly', so that $1_X$ is not the local unit but comes with a 2-cell $I \Rightarrow 1_X$.
If you're talking about strong monoidal functors then enriching over SymMonCat will by definition imply that identities are (isomorphic to) local units.
I don't know what you're trying to do exactly, but it seems it's just a matter of picking the right kind of functor, and thus the right category to enrich over.
A: Yes, the tensor product I want is the one described by Vincent Schmitt; it satisfies the universal property that any other braided monoidal bifunctor is naturally isomorphic to it.
