  We construct a novel class of spherically symmetric and asymptotically flat
black holes and naked singularities surrounded by anisotropic dark matter fluid
with the equation of state (EoS) of the form $P_t=\omega \rho$. We assume that
dark matter is made of weakly interacting particles orbiting around the
supermassive black hole in the galactic center and the dark matter halo is
formed by means of Einstein clusters having only tangential pressure. In the
large distance from the black hole, we obtain the constant flat curve with the
upper bound for the dark matter state parameter $\omega \lesssim 10^{-7}$. To
this end, we also check the energy conditions of the dark matter fluid outside
the black hole/naked singularity, the deflection of light by the galaxy, and
the shadow images of the Sgr A$^\star$ black hole using the rotating and
radiating particles. For the black hole case, we find that the effect of dark
matter fluid on the shadow radius is small, in particular the angular radius of
the black hole is shown to increase by the order $10^{-4} \mu$arcsec compared
to the vacuum solution. For the naked singularity we obtain significantly
smaller shadow radius compared to the black hole case. Finally, we study the
stability of the S2 star orbit around Sgr A$^\star$ black hole under dark
matter effects. It is argued that the motion of S2 star orbit is stable for
values $\omega \lesssim 10^{-7}$, however further increase of $\omega $ leads
to unstable orbits. Using the observational result for the shadow images of the
Sgr A$^\star$ reported by the EHT along with the tightest constraint for
$\omega$ found from the constant flat curve, we show that the black hole model
is consistent with the data while the naked singularity in our model can be
ruled out.
