  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 $p\in \{n-2,n-3,n-4\}$.
