  We compute explicitly Dirichlet generating functions enumerating
finite-dimensional irreducible complex representations of various $p$-adic
analytic and adelic profinite groups of type $\mathsf{A}_2$. This has
consequences for the representation zeta functions of arithmetic groups $\Gamma
\subset \mathbf{H}(k)$, where $k$ is a number field and $\mathbf{H}$ a $k$-form
of $\mathsf{SL}_3$: assuming that $\Gamma$ possesses the strong Congruence
Subgroup Property, we obtain precise, uniform estimates for the representation
growth of $\Gamma$. Our results are based on explicit, uniform formulae for the
representation zeta functions of the $p$-adic analytic groups
$\mathsf{SL}_3(\mathfrak{o})$ and $\mathsf{SU}_3(\mathfrak{o})$, where
$\mathfrak{o}$ is a compact discrete valuation ring of characteristic $0$.
These formulae build on our classification of similarity classes of integral
$\mathfrak{p}$-adic $3\times3$ matrices in $\mathfrak{gl}_3(\mathfrak{o})$ and
$\mathfrak{gu}_3(\mathfrak{o})$, where $\mathfrak{o}$ is a compact discrete
valuation ring of arbitrary characteristic. Organising the similarity classes
by invariants which we call their shadows allows us to combine the Kirillov
orbit method with Clifford theory to obtain explicit formulae for
representation zeta functions. In a different direction we introduce and
compute certain similarity class zeta functions. Our methods also yield
formulae for representation zeta functions of various finite subquotients of
groups of the form $\mathsf{SL}_3(\mathfrak{o})$,
$\mathsf{SU}_3(\mathfrak{o})$, $\mathsf{GL}_3(\mathfrak{o})$, and
$\mathsf{GU}_3(\mathfrak{o})$, arising from the respective congruence
filtrations; these formulae are valid in case that the characteristic of
$\mathfrak{o}$ is either $0$ or sufficiently large. Analysis of some of these
formulae leads us to observe $p$-adic analogues of `Ennola duality'.
