Arithmetic of random variables and bias-variance tradeoff derivation

So I'm a bit confused about random variables and products. Suppose we have two random variables $X,Y:\Omega \rightarrow \mathbb R$. Now here is something I don't quite get. In the case of $XY$ we think of it as a joint distribution. In other words $XY(a,b)=X(a)Y(b)$. But in the case of $X^2$ it seems we just think of it as the point wise product $X^2(a)=X(a)^2$ from some places I have seen. So why do we consider $XY:\Omega\times \Omega\rightarrow \mathbb{R}$ whereas $X^2:\Omega\rightarrow \mathbb R$. I apologize if this is trivial, but I don't have a background in probability theory, but I'm trying to understand the Bias-Variance proof below from Wikipedia. Namely, I don't understand why $E(f\hat{f})=E(f)E(\hat{f})$. When they take the product of the two random variables is it a point wise product or is it interpreted as a joint distribution?

https://en.wikipedia.org/wiki/Bias–variance_tradeoff#Derivation

EDIT: Basically I just want to clarify that whenever we take a product of two random variables its always interpreted as a joint distribution instead of point wise multiplication. Is this interpretation correct? If this is true then it answers my question since $f$ and $\hat{f}$ are independent random variables.