[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.