If there's a hyperbola of the form $$y = \frac{ax + b}{cx - a}$$ then the hyperbola and its inverse will always be same.
Can this be generalized more in any sense?
One obvious thing that could be done is we can multiply and divide $f(x)$ by another polynomial $g(x)$ and the result will still hold.
I don't know the exact type of generalization that you have in your mind. But since you said "in any sense", I will say a few things about the maximum possible generalization: Involutive Functions
An involution is a function $f: X\to X$ whose inverse is itself. That is, $$f\circ f=\operatorname{id}_X$$ A function is an involution if and only if the underlying relation is symmetric. When $X$ is a subset of $\mathbb{R},$ this fact can be seen as the graph of $f$ being symmetric around the line $y=x.$
The example gave you $$y=\dfrac{ax+b}{cx-a}$$ is an involution of $\mathbb{R}$ which can also extend easily to $\mathbb{R}\cup\{\infty\},$ or even to the Rienamm sphere $\mathbb{C}\cup\{\infty\}.$ And there are many more such functions. Another example is $$y=\dfrac{ax}{\sqrt{x^2-a^2}}.$$
Added Later:
Consider the function $a\mapsto 9a$ defined on $\mathbb{Z}_{20}$ (integers modulo $20$). It is not difficult to see that this is an involution. Likewise, in general, there are many exotic examples.
