Topology on the space of paths
Your equivalence relation seems well-suited to studying just the image of the path, so in a metric space, you could use the Hausdorff metric on the collection of images of the paths. The Hausdorff metric is where the distance between two compact sets $A,B$ is the supremum of all distances $d(a,B)$ and $d(b,A)$ for points $a$ in $A$ and $B$ in $B$.
In this topology, two equivalence slashes are close if their images are almost the same.
However, this doesn't work completely well since some paths have the same image without being equivalent.