Comment by frotaur

This is certainly an interesting question; however, it's probably very hard. This is because the 'set of all possible metrics of the 2d plane' is extremely large, and I am not sure if we have a good characterization for it.

There are a bunch of very strange metrics, e.g. a metric for which d(x,x) = 0, d(x,y)=1, that is, all points are at a distance of 1 to each other (this satisfies all axioms).