$ Ax\cdot x>0$ and $Ay\cdot y>0$ implies $(Ax\cdot x)(Ay\cdot y)\geq (Ax\cdot y)^2$?

For a counterexample, suppose we have $$A = \begin{bmatrix} 1 & 5 \\ 5 & 1 \end{bmatrix}, x = e_1 = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, y = e_2 = \begin{bmatrix} 0 \\ 1 \end{bmatrix}.$$ Then $Ax \cdot x = 1$ and $Ay \cdot y = 1$, but on the other hand, $Ax \cdot y = 5$.

On the other hand, the inequality is true if $A$ is a positive definite symmetric matrix (or even positive semidefinite). The idea here is: if $A$ is positive definite symmetric, then that implies $\langle x, y \rangle := Ax \cdot y$ forms an inner product on $\mathbb{R}^n$, and the Cauchy-Schwarz inequality for this inner product gives exactly the desired result.