We can think of these transformations from two perspectives: as mappings of the plane or as graphs of pre/post composition of the function.
For instance, the map $T: \mathbb{R}^2 \to \mathbb{R}^2$ given by $(a,b) \mapsto (a+2,b)$ is the translation of the plane two units to the right.
Let $\phi: \mathbb{R} \to \mathbb{R}$ be the function $\phi(x) = x-2$.
Let $f:\mathbb{R} \to \mathbb{R}$ be an arbitrary function, and let $\Gamma_f$ be the graph of this function.
Then it is a theorem that $T$ maps $\Gamma_f$ bijectively onto $\Gamma_{f \circ \phi}$. This theorem is usually stated in much less awesome language in precalculus textbooks.
In precalculus, we almost entirely restrict ourself to the second perspective of function pre/post composition. The first notion is almost entirely absent: at this level, we only treat real valued functions "of one variable", so mappings like $T$ are not defined or even thought of as functions.
In precalculus we do think about pre/post composition with functions other than affine functions. This happens every time we graph a composition of two functions like $\sin(x^2)$. It also happens with "log/log" plots. We just do not generally think of it as a "transformation" of the graph, because we do not have a notion of a mapping of the plane to itself. Even with the affine functions this is only done informally.
Pre/post composition with affine functions is an important special case with important properties. In particular, we can reason about these using the theorems and techniques of high school geometry (while we cannot similarly reason about more general mappings of the plane, requiring either linear algebra for more general affine mappings, or multivariable calculus for smooth mappings).
We do not generally consider other affine transformations because (as noted) we do not consider functions $\mathbb{R}^2 \to \mathbb{R}^2$ at all, if we did they do not correspond to function pre/post composition, they do not take graphs of functions to graphs of functions, and they require too much linear algebra to be understandable to students at this level. The one exception (as you note) is reflection in $y=x$, which is important because it corresponds to taking the inverse of a function.
Moreover there are extremely practical reasons to be concerned with pre/post composition with affine functions: they represent "affine coordinate" changes, which come up all the time in real life!
If $F(t)$ represents the temperature in Fahrenheit $t$ hours after noon, and $C(t)$ represents the temperature in Celsius $t$ minutes after 8:00am, then
$$F(t) = \frac{9}{5}C(60 \cdot (t+4)) + 32$$
or in other words
$$F = \phi_2 \circ C \circ \phi_1$$
Where $\phi_1(t) = 60 \cdot (t+4)$ is the conversion from "hours after noon" to "minutes after 8:00am" and $\phi_2(x) = \frac{9}{5}x+32$ is the conversion from Celsius to Fahrenheit.
Since affine changes of coordinates are so ubiquitous in the universe (and are indeed the basic building blocks of all other coordinate changes: this is called linear algebra and multivariable differential calculus), it makes sense to attend to the basics in high school!
