We might look at finding the best-fit line to a set of points as well as finding the shortest path between two nodes in a graph. It is clear we can abstract the idea of "optimization" from the two problems (we care about some optimal value, relative to a metric), but the ways in which we optimize the two are going to be very different.

I get what you mean, but in this instance both least square fit and shortest path can be written as a convex program and can be minimized using the same framework.