Given a graph $G$, the exact distance-$p$ graph $G^{[\natural p]}$ has $V(G)$ as its vertex set, and two vertices are adjacent whenever the distance between them in $G$ equals $p$. We present formulas describing the structure of exact distance-$p$ graphs of the Cartesian, the strong, and the lexicographic product. We prove such formulas for the exact distance-$2$ graphs of direct products of graphs. We also consider infinite grids and some other product structures. We characterize the products of graphs of which exact distance graphs are connected. The exact distance-$p$ graphs of hypercubes $Q_n$ are also studied. As these graphs contain generalized Johnson graphs as induced subgraphs, we use some known and find some new constructions of their colorings. These constructions are applied for colorings of the exact distance-$p$ graphs of hypercubes with the focus on the chromatic number of $Q_{n}^{[\natural p]}$ for $p\in \{n-2,n-3,n-4\}$.