Is there a name for metric spaces where the path for shortest distance is unique? In normal Euclidean space with the $L_2$ metric, the shortest path between two points is a straight and unique line. However, on the taxi-cab metric ($L_1$), between any two points that do not lie on the same vertical or horizontal line, there are an infinite number of shortest paths between all with the same path distance.

Is there a name for and/or a way determine whether a given metric has unique shortest paths?
you might be interested in Riemannian manifolds $(M,d)$, which are manifolds that come along with a metric induced by $d$. 
This metric then helps to find (locally) shortest paths between points on that manifolds. These paths are called geodesics and are solutions of ODE's equations. In Euclidean (flat) spaces, they are just straight lines, but e.g. on a sphere, they are given by the great circles.
In other words, if you can give your object a smooth manifold structure, you can find unique shortest paths. Hope this helps.
you might be interested in Riemannian manifolds $(M,d)$, which are manifolds that come along with a metric induced by $d$. 
This metric then helps to find (locally) shortest paths between points on that manifolds. These paths are called geodesics and are solutions of ODE's equations. In Euclidean (flat) spaces, they are just straight lines, but e.g. on a sphere, they are given by the great circles.
In other words, if you can give your object a smooth manifold structure, you can find unique shortest paths. Hope this helps.