  All black holes (BHs) in nature are expected to be described by the Kerr
vacuum solution of general relativity. However, the Kerr solution comes with
several difficulties such as the existence of Cauchy horizons, curvature
singularities, and causality-violating regions. Attempts to resolve some of
these issues include phenomenological BH models, which typically contain
nontrivial matter content. We introduce a simple framework here to examine the
properties of matter in such phenomenological models for a broad class of
stationary and axisymmetric spinning BH spacetimes, generated from nonspinning
seed solutions via a metric ansatz. We apply this framework to a representative
set of spinning BH spacetimes and the non-spinning seeds from which they are
derived. The models span different types of matter - fluids, scalar fields,
electromagnetic fields. For each model, we calculate the timelike four-velocity
of the matter and thereby identify the rest frame of the matter, both outside
and inside the horizon. We then examine the spatial distribution of the matter
rest-frame energy density $\epsilon$ and the principal pressures. This provides
a complete picture of how the matter moves, what its material properties are,
and whether it obeys the classical energy conditions. Notably, at a horizon,
the normal component of the pressure always satisfies $p_n = -\epsilon$. We
also investigate the expansions of the principal null congruences and explore
the Hawking mass profiles of these spacetimes. These provide glimpses into the
geometry of the stationary BH exterior as well as the nonstationary interior
cosmology. The axisymmetric metric ansatz we work with can be used to generate
new spinning solutions from a variety of nonspinning seeds. The matter in these
models often satisfies the weak energy condition, at least in the BH exterior,
and some models exhibit non-rigid, differential rotation.
