Neutron stars may exhibit pressure anisotropy arising from several microphysical processes, including strong magnetic fields, elastic stresses, and viscous effects. In this work, we investigate how pressure anisotropy influences two key observables: the tidal deformability measured in gravitational-wave signals from binary inspirals, and the fundamental quadrupolar mode (f-mode) oscillation frequency associated with stellar perturbations. We model anisotropic neutron stars using a phenomenological quasi-local prescription governed by a single dimensionless anisotropy parameter, allowing us to systematically explore deviations from isotropy. Our analysis demonstrates that, although the universal relation between tidal deformability and f-mode frequency varies with the strength of anisotropy, it remains remarkably insensitive to changes in the underlying equation of state that relates radial pressure to energy density. This behavior mirrors the well-known approximate universality found in isotropic stars. Taking advantage of this anisotropy-dependent universal relation, we perform Bayesian inference to constrain the anisotropy parameter using data from both the gravitational-wave event GW170817 and a simulated GW170817-like event detected by a future network. We show that current data restrict the anisotropy parameter to values of order unity, and that future measurements yield comparable bounds. Notably, these constraints are only weakly influenced by remaining uncertainties in the neutron-star equation of state.