  BaCo$_2$(AsO$_4$)$_2$ (BCAO), a honeycomb cobaltate, is considered a
promising candidate for materials displaying the Kitaev quantum spin liquid
state. This assumption is based on the distinctive characteristics of Co$^{2+}$
ions (3$d^7$) within an octahedral crystal environment, resulting in
spin-orbit-coupled $J_{\rm eff}$~=~1/2 doublet states. However, recent
experimental observations and theoretical analyses have raised questions
regarding this hypothesis. Despite these uncertainties, reports of continuum
excitations reminiscent of spinon excitations have prompted further
investigations. In this study, we explore the magnetic phases of BCAO under
both in-plane and out-of-plane magnetic fields, employing dc and ac magnetic
susceptibilities, capacitance, and torque magnetometry measurement. Our results
affirm the existence of multiple field-induced magnetic phases, with strong
anisotropy of the phase boundaries between in-plane and out-of-plane fields. To
elucidate the nature of these phases, we develop a minimal anisotropic exchange
model. This model, supported by combined first principles calculations and
theoretical modeling, quantitatively reproduces our experimental data. In BCAO,
the combination of strong bond-independent XXZ anisotropy and geometric
frustration leads to significant quantum order by disorder effects that
stabilize collinear phases under both zero and finite magnetic fields.
