[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?

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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.