The exceptional isomorphism between PGL(3,2) and PSL(2,7): geometric origin?

There is an explanation of sorts in Section 1.4 of Elkies's "The Klein quartic in number theory". There is a three-dimensional lattice $L$ over the cyclotomic field $k=\mathbf Q(\zeta_7)$, and $G$ can be defined as its group of isometries. The resulting three-dimensional representation of $G$ has the unusual property of remaining irreducible when reduced modulo every prime of $\mathcal O_k$. Its reduction modulo a prime over $2$ turns out to be $\mathrm{GL}(3,\mathbf F_2)$ acting on $\mathbf F_2^3$, and its reduction modulo a prime over $7$ is $\mathrm{PSL}(2,\mathbf F_7)$ acting on $\mathbf F_7^3$ as the symmetric square of the two-dimensional representation of $\mathrm{SL}(2,\mathbf F_7)$. (Note that since $-1$ acts trivially on the symmetric square, the symmetric square really is a projective representation.)

V. Dotsenko's construction, on math.stackexchange:

https://math.stackexchange.com/questions/1401/why-psl-3-mathbb-f-2-cong-psl-2-mathbb-f-7/1450#1450

may fit your requirement "combinatorial mapping of these geometries that induces an isomorphism".