Does the intersection of sets have a categorical interpretation? My question is the title, really. I am wondering if the intersection of sets can be seen as a categorical construction on the objects of $\mathbf{Set}$.

No, the intersection of two "isolated" sets $A,B$ doesn't have any categorical interpretation. Because for any reasonable meaning of this, we would like to have $A \cap B \cong A' \cap B'$ if $A \cong A'$ and $B \cong B'$. But this is clearly wrong (take $A=B=A'=\{1\}$ and $B'=\{0\}$).
Thus, from the perspective of category theory, the set-theoretic operation $\cap$ doesn't make much sense. What is $\pi \cap \mathbb{R}$ supposed to be? However, it is meaningful to take the intersection with respect to two (injective) maps $A \to C$ and $B \to C$. Namely, then the pullback $A \times_C B$ is the desired intersection. For more on this, see also math.SE/295800 and math.SE/704593 and math.SE/866127.
