According to this: link we can allow the $\emptyset$ to be a metric space. But by definition what will the metric be? If $(\emptyset,d)$ is a metric space, what must $d$ be?
It is supposed to be a function such that $d: \emptyset\times \emptyset \rightarrow \mathbb{R}$, but this means that it is a funciton such that $d: \emptyset \rightarrow \mathbb{R}.$
Does this mean that it can be any function function taking values in $\mathbb{R}$? Or is $d=\emptyset$?, so the metric space is $(\emptyset,\emptyset)?$
The distance function is indeed the function $d:\emptyset \to \mathbb{R}$. It is the function $d=\emptyset$, as you said. Because it has no elements in its domain, it has an empty image. However, it can have any codomain you want, it just will not be surjective unless that codomain is empty.
Note: $d$ is a function mapping $\emptyset \times \emptyset\to \mathbb{R}$, because $\emptyset \times \emptyset = \emptyset$ (This is true because that cartesian product is defined as the set of all ordered pairs made from elements of $\emptyset$ and $\emptyset$. That set is empty).