  Let $q$ be an odd power of a prime $p\in \mathbb{N}$, and
$\mathrm{PPSP}(\sqrt{q})$ be the finite set of isomorphism classes of
principally polarized superspecial abelian surfaces in the simple isogeny class
over $\mathbb{F}_q$ corresponding to the real Weil $q$-numbers $\pm \sqrt{q}$.
We produce explicit formulas for $\mathrm{PPSP}(\sqrt{q})$ of the following
three kinds: (i) the class number formula, i.e. the cardinality of
$\mathrm{PPSP}(\sqrt{q})$; (ii) the type number formula, i.e. the number of
endomorphism rings up to isomorphism of members of $\mathrm{PPSP}(\sqrt{q})$;
(iii) the refined class number formula with respect to each finite group
$\mathbf{G}$, i.e. the number of elements of $\mathrm{PPSP}(\sqrt{q})$ whose
automorphism group coincides with $\mathbf{G}$. Similar formulas are obtained
for other polarized superspecial members of this isogeny class using
polarization modules. We observe several surprising identities involving the
arithmetic genus of certain Hilbert modular surface on one side and the class
number or type number of $(P, P_+)$-polarized superspecial abelian surfaces in
this isogeny class on the other side.
