Does $\frac{d x}{dy}$ indicate that $x=f(y)$? Does the expression $\dfrac{d x}{dy}$ indicate that $x=f(y)$ ? Can we compute it even if $y=g(x)$ ?
For example, if $y=g(x)=x^2,$ is $\dfrac{d x}{dy}$ meaningful even though $y=g(x),$ rather than $x=f(y)$ ?

$y=g(x)$ establishes a functional relation between $x$ and $y$, and under some conditions, it can be inverted as $x=g^{-1}(y)$. Hence there is no real difference between $y=g(x)$ and $x=f(y)$.
This said, when the derivatives exist, the following relation holds:
$$\frac{dx}{dy}=\frac1{\dfrac{dy}{dx}}.$$

Illustration:
For positive $x$ and $y$,
$$y=x^2\iff x=\sqrt y.$$
Then using the common rules of differentiation,
$$\frac{dy}{dx}=2x$$
and
$$\frac{dx}{dy}=\frac1{2\sqrt y}=\frac1{2x}.$$
