The CLASP2 (Chromospheric LAyer Spectro-Polarimeter 2) sounding rocket mission was launched on 2019 April 11. CLASP2 measured the four Stokes parameters of the Mg ii h and k spectral region around 2800 Å along a 200'' slit at three locations on the solar disk, achieving the first spatially and spectrally resolved observations of the solar polarization in this near-ultraviolet region. The focus of the work presented here is the center-to-limb variation of the linear polarization across these resonance lines, which is produced by the scattering of anisotropic radiation in the solar atmosphere. The linear polarization signals of the Mg ii h and k lines are sensitive to the magnetic field from the low to the upper chromosphere through the Hanle and magneto-optical effects. We compare the observations to theoretical predictions from radiative transfer calculations in unmagnetized semiempirical models, arguing that magnetic fields and horizontal inhomogeneities are needed to explain the observed polarization signals and spatial variations. This comparison is an important step in both validating and refining our understanding of the physical origin of these polarization signatures, and also in paving the way toward future space telescopes for probing the magnetic fields of the solar upper atmosphere via ultraviolet spectropolarimetry.