Fuzzy mathematics is traditionally done from the logical viewpoint, so the first step in introducing fuzzy metrics is often the transformation f(x,y) = exp(-d(x,y)).
Then we have the following correspondences:
d(x,y) = 0 if and only if f(x,y) = 1.
d(x,y) is plus infinity if and only if f(x,y) = 0.
d(x1, y1) < d(x2, y2) if and only if f(x1, y1) > f(x2, y2).
The axiom d(x,x) = 0 becomes f(x,x) = 1.
The axiom d(x,z) < d(x,y) + d(y,z) becomes f(x,y) * f(y,z) < f(x,z).
Non-expansive maps become maps which respect overlap by not letting it decrease.
However, this seems to be a rather superficial duality: basically two equivalent ways to write the same things using different notation.
The question is whether there is also a natural deeper duality here (of a contravariant nature, where function arrows would reverse direction when one switches between these two viewpoints).