Graphs with a polynomial number of shortest paths between any pair of vertices Let $G$ be a simple undirected graph, and let $s$ and $t$ be two arbitrary vertices of $G$. Even for some rather restricted graph classes, the number of shortest paths between $s$ and $t$ can be exponential in the size of $G$. However, for some graph classes like block graphs, the number of shortest paths between $s$ and $t$ is 1 (such graphs are known as geodetic graphs). 

For what graph classes $G$ is it true that between any pair of vertices $s$ and $t$ of $G$, the number of shortest paths between $s$ and $t$ is bounded from above by $\text{poly}(n,m)$? Here, $n$ is the number of vertices, and $m$ the number of edges.

For example, are there graphs (other than block graphs) that would have a constant number of shortest paths between any pair of vertices?

For your first question, you might be interested in the class of strongly regular graphs.
For your second question, cycles of odd length work since any two vertices have a unique shortest path between them. Actually any such (simple) graph would have to be either edgeless or geodetic since adjacent vertices have a unique shortest path between them. 
