  By employing modified Chaplygin fluid prescription for the dark energy, we
construct slowly rotating isotropic and anisotropic dark energy stars. The slow
rotation is incorporated via general relativistic Hartle-Thorne formalism;
whereas the anisotropy is introduced through Bowers-Liang prescription. We
consider both the monopole and quadrupole deformations and present a complete
analysis of rotating dark energy stars. By numerically solving the rotating
stellar structure equations in presence of anisotropy, we analyse and quantify
various properties of dark energy stars such as mass ($M$), radius, mass
deformation, angular momentum ($J$), moment of inertia, and quadrupole moment
($Q$), for three different equation of state parameters. We find that
anisotropic slow rotation results in significant deformation of stellar mass
and thereby affects other global properties studied. For the values of angular
frequencies considered, the effect of anisotropy on the stellar structure is
found to be more prominent than that due to rotation. The dimensionless
quadrupole moment $QM/J^2$ measuring deviation from a Kerr metric black hole
was obtained for anisotropic dark energy stars. We observe that dark energy
stars with higher anisotropic strength tend to approach the Kerr solution more
closely. We report that our results have considerable agreement with various
astrophysical observational measurements.
