It seems to me an equation, in an abstract sense, must always involve some varying quantities where the varying quantities belong in some space (set, algebraic structure, what have you). In order to make precise the phrase, "vary quantities", it seems to me that one must have a mechanism for evaluation of each side of the equation. Ultimately, I think solving an equation must always be, in essence, finding the pre-image of a mapping. If the pre-image is empty then there are no solutions.

Consider the equation over $\mathbb{C}$ $$x^2-3x+2=0$$

We have that $x^2-3x+2$ is a polynomial and this polynomial induces a natural map from to $\mathbb{C}$ to $\mathbb{C}$ called evaluation. The equation is really asking for the pre-image of $0$ of this map.

Consider the functional equation $$f(x+y) + f(x-y) = 0$$ where we are looking for solutions that are functions from $\mathbb{R}$ to $\mathbb{R}$. I think ultimately that this equation can be thought of in terms of the pre-image of the map $G$ that takes functions like $f$ and maps them to the function from $\mathbb{R^2}$ to $\mathbb{R}$ by sending $f$ to the function of two variables $f(x+y)+f(x-y)$. And the equation is really asking for the pre-image of the zero function under this map.

Is it correct to view all equations in this manner? That is finding the solutions must always be equivalent to finding the pre-image of some element of some mapping?

Think of basic equations one finds in college algebra books. I tell my students that we take the given equation and apply solution preserving operations to it to transform the equation into a simpler one. The goal is to ultimately end up with a simpler equation whose solutions we can find by inspection. For instance,

$$3x - 2 = 5 \implies 3x = 7 \implies x = \frac{7}{3}$$

At each step we transform the equation to a simpler one whose solution set is the same. We can solve the last equation by inspection. Ultimately, isn't this how all equations are solved? We transform the equation to simpler equation(s) and end up with an equation that can be solved by inspection.