  We investigate magnetic field $(eB)$ dependence of constituent quark mass,
the longitudinal and transverse pressure as well as the magnetization and
magnetic susceptibility of strongly interacting quark matter. We employ the
two-flavour Polyakov Nambu--Jona-Lasinio model with the inclusion of the
anomalous magnetic moment (AMM) of the quarks at finite temperature $(T)$ and
finite quark chemical potential $(\mu_q)$ capturing different stages of chiral
phase transition. We find that the transverse pressure, magnetization and
magnetic susceptibility become highly oscillatory for large values of $eB$ in
the chiral symmetry broken phase. However the oscillations cease to occur at
higher values of $T$ and $\mu_q$ when chiral symmetry is (partially) restored
and the anisotropic nature of the pressure becomes significant even at smaller
values of $eB$. As the inclusion of AMM of the quarks leads to inverse magnetic
catalysis of the transition temperature we observe that the variations of
transverse pressure, magnetization and magnetic susceptibility are
significantly modified in the vicinity of the chiral transition temperature.
Furthermore, above the chiral transition temperature the magnetic
susceptibility is found to remain positive for a wide range of $eB$ indicating
a paramagnetic character of the strongly interacting quark matter. Finally, we
have also examined the magnetism of strongly interacting matter in the
quarkyonic phase. The obtained results could be useful for a
magnetohydrodynamic evolution of hot and dense matter created in heavy-ion
collisions.
