Suppose I have two different scalar products on two spaces $X, \langle \cdot,\cdot\rangle_X$ and $Y, \langle \cdot,\cdot\rangle_Y$. Given points $x_i\in X$ and $y_i\in Y$ is the following valid?
$$\sum_{i,j}^n\langle x_i,x_j\rangle_X \langle y_i,y_j\rangle_Y= \langle\sum_{i}^n x_i,\sum_{j}^n x_j\rangle_X \langle\sum_{i}^n y_i,\sum_{j}^ny_j\rangle_Y$$
If so the latter would be equal $\|\sum_{i}^nx_i\|^2\|\sum_{i}^ny_i\|^2$ which would be handy for me calculation
No.
Suppose $X=Y=\mathbb R$ and all $x_i, y_i$ are $1$.
Then your proposed identity claims that $n^2=n^4$.