Thinking about – and inspired by – an insightful example by Aaron Meyerowitz I found a convincing argument that there is probably no generally agreed upon concept of distance-likeness involving a sensibly generalized triangle inequality. The argument goes like this:
Consider a world of non-intersecting disks in the plane. Let the distance $d$ between two disks be the smallest Euclidean distance between any two points on their respective boundaries. This involves that any two touching disks have distance 0 and that the distance-law d(x,y)=0 iff x=y is violated. I assume that it is nevertheless generally agreed upon that this is a sensible distance.
But any sensible triangle inequality is violated, too. That's because for every two non-touching disks $x,z$ that touch a common disk $y$ we have
$$d(x,z) > d(x,y) + d(y,z) = 0$$
and $d(x,z)$ can be arbitrarily large (depending on the radius of disk $y$) and thus greater than $F(0,0)$ for any function $F(u,v)$.
The questions question arises for what specific reasons we nevertheless do believe that $d$ is a sensible distance?
Note, that for a given maximal disk radius $r$max we may get a sensible version of the triangle inequality (due to Aaron's example) and with $r$max $\rightarrow 0$ we get the usual triangle inequality.