I am not sure that what I am doing has any mathematical basis, but would like some help to understand if what I am doing actually even makes sense mathematically as (maybe unintentionally) the result is very useful to me.
Consider that you have the space of polynomials up to (and including) degree $d$ in $n$ variables (which I have denoted $\mathbb{R}_d[x_1,\dotsc,x_n]$) and you have two inner products defined on the space, for $u,v\in\mathbb{R}_d[x_1,\dotsc,x_n]$: \begin{align} \left<u,v\right> &= \idotsint_{x_1,\dotsc,x_n\in[0,1]}u\cdot v\,\mathrm{d}x_1\dotsb\mathrm{d}x_n \\ \left[u,v\right] &= \idotsint_{x_1,\dotsc,x_n\in[0,2]}u\cdot v\,\mathrm{d}x_1\dotsb\mathrm{d}x_n. \end{align}
If I have a basis $\beta=\{\beta_1,\dotsc,\beta_k\}$ (with some appropriate $k$) for my space then for each pair of basis elements in $\beta$, I generate the matrices $A_{ij}=\left<\beta_i,\beta_j\right>$ and $B_{ij}=\left[\beta_i,\beta_j\right]$ is there any meaning behind the matrix $C=A-B$?
Well, we have $$C_{ij} = -\int_{[1,2]^n} \beta_i\cdot \beta_j\,d\mathbf{x}$$
so the matrix $-C$ is the generated by the inner product $$(u,v) \mapsto \int_{[1,2]^n} u\cdot v\,d\mathbf{x}$$ from the basis $\beta$ in the same way as $A$ and $B$.