If $f(x) = f(-x),$ then the function is said to be symmetric, so the transformation $x \rightarrow -x$ is an example of a symmetry.
Trig functions like $\sin(x)$ and $\cos(x)$ satisfy the property that $f(x + 2\pi) = f(x)$ and so their manifold remains invariant under shifting. I don't know if this is technically a "symmetry" or not, but if it is, I'd classify it as a shift symmetry.
Is there a name for if/when $f(ax) = f(x),$ for some fixed $a \in \mathbb{R}$?
More generally, is there an area of math that studies functional relationships of the form $f(g(x)) = f(x)$?
A symmetry of the form $f(ax) = f(x)$ is called a scaling symmetry, or sometimes a dilation or dilatation; generally, a function satisfying $f(ax) = a^k f(x)$ for all scalars $a$ and some $k \in \mathbb{Z}$ is called homogeneous of degree $k$.
The study of symmetry is called group theory and the study of functions invariant under groups of symmetries is called invariant theory. This is an old classical branch of mathematics and many things are known. Closely related are representation theory and harmonic analysis, although really the basic idea is so fundamental it appears in all sorts of other places as well. For example modular forms are certain special functions satisfying an almost-symmetry with respect to a group called the modular group, and they are quite important in number theory.