What is a term for a value $x \in X$ that for binary operator $f\colon X\times X \to X$ maps always to itself, no matter what is the other value $$ \forall y \in X \quad f(x, y) = f(y, x) = x $$
Is it called "fixed point" too or not?
Examples are $0$ for numbers multiplication, $\mathrm{false}$ for conjunction, $\mathrm{true}$ for disjunction, $\emptyset$ for intersection, etc.
Such elements are often called absorbing elements. In semigroup theory it is often called the identity or zero element.
see: https://en.wikipedia.org/wiki/Absorbing_element https://www.encyclopediaofmath.org/index.php/Absorbing_element https://planetmath.org/absorbingelement