Approximability of the edge-disjoint shortest paths problem
Below, we will assume there is exactly one shortest path between every pair of vertices. The following will also work for the case where $O(\log n)$ pairs of vertices have $O(1)$ shortest paths between them.

For each pair $\{s_i , t_i\}$ let the shortest path be $P_i$, $\forall i \in \{0,1\ldots k\}$. The shortest path can be found in polynomial time. Let us number the edges $e_1, e_2, \ldots e_q$. Let the maximum number of pairs $\{s_i , t_i\}$ for which there exist disjoint-shortest paths be $\text{OPT}$. 

Now, we reduce this problem to an optimization problem.

Define a matrix $A_{k\times q}$ such that $A_{ij} = 1 \text{ if } e_j \in P_i$.

$$
\text{OPT} = \max \sum_{i=1}^k x_i \quad 
\\
\text{s.t } Ax \leq \mathbf{1}_{n\times 1} \\ 
x \in \{0,1\}
$$

Let the optimal fractional solution for the above problem be $\text{OPT}_{\text{LP}}$ and $\text{OPT}_{\text{LP}} \geq \text{OPT}$. The above LP has an integrality gap of $O(\log k)$. Therefore, $O(\log k) \cdot \text{OPT}_{\text{IP}} \geq \text{OPT}_{\text{LP}}$. Thus $ O(\log k) \cdot \text{OPT}_{\text{IP}} \geq \text{OPT}$. Hence we obtain a $O(\log k)$ approximation algorithm.

The proof for the integrality gap result can be found in Vazirani's "Approximation Algorithms", chapter 20. Multicut in General Graphs.