I know it's not the question, but, anent now-user76479's / I guess-then Arul's comment and your response, one way to make use of an algebra structure on $W$, if it is present, is to define $V^{\otimes W} = V^{\otimes\operatorname{Hom}_\text{alg}(W, k)}$ (i.e., one ‘factor’ in the tensor product for each element of the set of algebra homomorphisms), where $k$ is the underlying field. Of course this can fail to exhibit the desired dimension behaviour for a randomly chosen finite-dimensional $k$-algebra $W$, but, if $W$ is an étale algebra, then it works as desired.

I know it's not the question, but, anent now-user76479's / I guess-then Arul's comment and your response, one way to make use of an algebra structure on $W$, if it is present, is to define $V^{\otimes W} = (V^{\otimes\operatorname{Hom}_\text{$k^\text{sep}$-alg}(W, k^\text{sep})})^{\operatorname{Gal}(k^\text{sep}/k)}$ (i.e., one ‘factor’ in the tensor product for each element of the set of algebra homomorphisms), where $k$ is the underlying field and $k^\text{sep}/k$ is a separable closure. Of course this can fail to exhibit the desired dimension behaviour for a randomly chosen finite-dimensional $k$-algebra $W$, but, if $W$ is an étale algebra, then it works as desired.

I know it's not the question, but, anent now-user76479's / I guess-then Arul's comment and your response, one way to make use of an algebra structure on $W$, if it is present, is to define $V^{\otimes W} = V^{\otimes\operatorname{Hom}_\text{alg}(W, k)}$ (i.e., one ‘factor’ in the tensor product for each element of the set of algebra homomorphisms), where $k$ is the underlying field. Of course this can fail for many interesting $W$, but, if $W$ is an étale algebra, then it works as desired.

I know it's not the question, but, anent now-user76479's / I guess-then Arul's comment and your response, one way to make use of an algebra structure on $W$, if it is present, is to define $V^{\otimes W} = V^{\otimes\operatorname{Hom}_\text{alg}(W, k)}$ (i.e., one ‘factor’ in the tensor product for each element of the set of algebra homomorphisms), where $k$ is the underlying field. Of course this can fail to exhibit the desired dimension behaviour for a randomly chosen finite-dimensional $k$-algebra $W$, but, if $W$ is an étale algebra, then it works as desired.