  Recently, non-zero rotation angle $\beta=0.30^\circ\pm0.11^\circ$
$(68\%\text{ C.L.})$ [Phys. Rev. Lett. \textbf{128}, no.9, 091302 (2022)] has
been reported for linear polarization of cosmic microwave background (CMB)
radiation, which is known as cosmic birefringence (CB). We used this
birefringence angle of CMB to study and distinguish different candidates of
dark matter (DM), e.g., dipolar and sterile neutrino DM. We calculated CMB
forward scattering by those probable candidates of DM to generate $\beta$ in
the presence of primordial scalar fluctuations' background. We explicitly
plotted bounds on the mass and electromagnetic coupling for different sectors
of DM, sterile neutrino, and dipolar DM, and compared them with other
experimental bounds. Regarding dipolar DM, our calculations put a bound on the
Majorana magnetic dipole moment about $\mathcal{M}\leqslant
1.4\times10^{-14}\,\frac{\beta}{0.30^\circ}\sqrt{\frac{m_{\text{\tiny{DM}}}}{1\,GeV}}\,
e.\text{\,cm}$. In the case of sterile neutrino DM, the bound on the mass and
mixing angle was estimated at $\theta^2 \leqslant
3.3\,(rad)^2\frac{\beta}{0.30^\circ}\,\frac{m_{DM}}{\rm{KeV}} $, which can be a
new constraint for sterile neutrino DM whose production mechanism is motivated
by models with a hidden sector coupled to the sterile neutrino. Based on our
results, if the constraint on the mass and the electromagnetic coupling for DM
must be within the allowed region, none of the considered candidates can
compensate for all the observed CB angles. We also discussed the maximum
contribution of the CB angle via CMB forward scattering by different sectors of
the dark matter.
