In this question
Derivative of a function with respect to another function.
the chain rule is stated in order to take derivative of single variable function with respect to a function of the same single variable.
What is the generalization of this rule to
$$\frac{\partial f(x,y)}{\partial g(x,y)}$$
or more variables?
The formalism in your question, $\frac{\partial f(x,y)}{\partial g(x,y)}$, does not make any sense. Partial differential operator $\partial$ on a function $f(x,y)$, by definition, gives you the partial derivative with respect to a single independent variable, not a whole function.
Suppose you have functions $f(x,y)$, $x(u,t)$, and $y(u,t)$. However, you want the partial derivative of $f(x,y)$ with respect to $u$, and not $t$. Then,
$$\frac{\partial f}{\partial u}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial u}$$
Regardless, if you notice, the partial differential operator does not differentiate function with respect to other functions, but a single variable. However, you can say that the variable itself is a function, but even then, we assume the variable to be independent, if not, then we would have to use the partial differential operator. Therefore, whenever we differentiate a function it is with respect to an independent variable and not a general function.
Let $f$ be a function of $x$ such that $f(x)$, and let $x$ be a function of $u$ and $v$ such that $x(u,v)$. Now notice differentiating $f$ with respect to $x$ is just:
$$\frac{df}{dx }$$
When differentiating $f$ with respect $x$, we assumed $x$ is the only argument of $f$, and therefore independent. On the other hand if we assume $x$ is not independent, then we will have to differentiate it with respect to the arguments of $x$ that are independent:
$$\frac{\partial f}{\partial u}=\frac{df}{dx}\frac{\partial x}{\partial u}$$And,$$\frac {\partial f}{\partial v}=\frac{df}{dx}\frac{\partial x}{\partial v}$$
Notice the difference in notation. Finally, regardless of how you differentiate, you always assume that you are differentiating with respect to a single independent variable and not a whole function. Therefore,
$$\frac{\partial f(x,y)}{\partial g(x,y)}$$ is not a correct symbol.
However, what you mean, as a generalization, is this: suppose $f$ is a function of $x$ and $y$. Furthermore, $x$ and $y$ are functions of $u$ and $v$ such that $x(u,v)$ and $y(u,v)$. Now, I can only take the derivative of $f$ wrt. $u$ or $v$. Let us take u for instance:
$$\frac{\partial f}{\partial u}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial u}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial u}$$However, if you notice that is not the complete derivative of $f$, but it is just the derivative of $f$ wrt. a single variable u.
That is why in order to get a relatively fuller derivative of $f$, we define directional derivative at a point (p,q) such that $D_f(p,q)$ in the direction of $\vec v$ is defined as:$$f_x(p,q)\vec i+ f_y(p,q)\vec j\cdot \vec v$$
