It seems that in topology, we have very good ways to deduce whether a space is metrizabe or not.
For example, if a space $(X, \mathcal{T})$ is metrizable, then it is Hausdorff, $T_3$, $T_4$, $\ldots$, $T_6$, first countable, separable $\Leftrightarrow$ second countable, and we can do even further by applying metrizability theorems like Urysohn's theorem.
Then by negation, if we are missing some of those properties, then it is not metrizable.
But how do people come up with the actual metric on those spaces? It seems to me as far a $\mathbb{R}$ is concerned, there is sort of this intuitive/common sense feeling what the metric should be. For example, the absolute value of distance between two points. And lo and behold, it just happens to satisfy the metric axioms.
So to me it is not very clear if this way will always work for weirder topological spaces that happens to be metrizable. Are there metrizable spaces for which we do not know what metric actually generates the topology?
There is no such thing as "the" metric. There are lots of equivalent metrics that will generate the same topology. How you might construct one might have a lot to do with how you know the space is metrizable.