XOR and bilinear form property
Actually, as you note, XOR doesn't satisfy the bilinear condition.

What does is multiplication $\otimes$ in $\mathbb{F}_2$, that is:

$$\otimes(u \oplus v, w) = \otimes(u,w) \oplus \otimes(v,w)$$

or, as more traditionally expressed (using $+$ and $\times$ as the field operations):

$$(u+v) \times w = (u \times w) + (v \times w)$$

This is one of the fundamental properties of a field; since $GF(2)$ is a field (with $\oplus$ and $\otimes$ as the field addition and field multiplication), this holds.

And, in $GF(2)$, $\otimes$ is equivalent to bitwise $AND$