We report spin-polarized scanning tunneling microscopy measurements of an Anderson impurity system in MoS_{2} mirror-twin boundaries, where both the quantum-confined impurity state and the Kondo resonance resulting from the interaction with the substrate are accessible. Using a spin-polarized tip, we observe magnetic-field-induced changes in the peak heights of the Anderson impurity states as well as in the magnetic-field-split Kondo resonance. Quantitative comparison with numerical renormalization group calculations provides evidence of the notable spin polarization of the spin-resolved impurity spectral function under the influence of a magnetic field. Moreover, we extract the field and temperature dependence of the impurity magnetization from the differential conductance measurements and demonstrate that this exhibits the universality and asymptotic freedom of the S=1/2 Kondo effect. This work shows that mirror-twin boundaries can be used as a testing ground for theoretical predictions on quantum impurity models.