Proposition: For any nonsingular $A\in\mathbb{C^{n\times n}}$ and nonsingular $B\in\mathbb{C^{n\times n}}$, the following holds
$$\frac{\Re(x_i^*Ax_i)}{\sigma_i(A)}=\frac{\Re(y_j^*B^*ABy_j)}{\sigma_j(B^*AB)} \quad \text{for }i,j=1,2,\ldots,n.$$
where $[\sigma_1(A)\quad\sigma_2(A)\quad\cdots\quad\sigma_n(A)]$ are singular values of $A$, and $x_i(A)$ are corresponding singular vectors. Similarly $y_i(B^*AB)$ are corresponding singular vectors to $\sigma_i(B^*AB)$, where $[\sigma_1(B^*AB)\quad\sigma_2(B^*AB)\quad\cdots\quad\sigma_n(B^*AB)]$ are singular values of $B^*AB$.
Here $x_i$ are just columns of $V$ in svd of $A$, i.e., $A=U\Sigma V^*.$
I conjecture this after working with some engineering problem. Few simulations show that it holds for the examples I have tried. It is surprising for me that this proposition works even for one random example, but at the same time I was unable to prove it, so decided to share.
EDIT. I just found counter-example when $A,B$ random random nonsingular matrices. Turns out the matrices I used before were $A$ normal and $B$ positive definite.