Young brown dwarfs are analogs to giant exoplanets, as they share effective temperatures, near-infrared colors and surface gravities. Thus, the detailed characterization of young brown dwarfs might shed light on the study of giant exoplanets, that we are currently unable to observe with the sufficient signal-to-noise to allow a precise characterization of their atmospheres. 2MASS J22081363+2921215 is a young L3 brown dwarf, member of the β-Pictoris young moving group (23 ± 3 Myr), that shares its effective temperature and mass with the AB Pictoris b giant exoplanet. We performed a ~2.5 hr spectro-photometric J-band monitoring of 2MASS J22081363+2921215 with the MOSFIRE multi-object spectrograph, installed at the Keck I telescope. We measured a minimum variability amplitude of 3.22 ± 0.42% for its J-band light curve. The ratio between the maximum and the minimum flux spectra of 2MASS J22081363+2921215 shows a weak wavelength dependence, and a potential enhanced variability amplitude in the alkali lines. Further analysis suggests that the variability amplitude on the alkali lines is higher than the overall variability amplitude (4.5–11%, depending on the lines). The variability amplitudes in these lines are lower if we degrade the resolution of the original MOSFIRE spectra to R ~ 100, which explains why this potential enhanced variability in the alkali lines has not been found previously in HST/WFC3 light curves. Using radiative-transfer models, we obtained the different cloud layers that might be introducing the spectro-photometric variability we observe for 2MASS J22081363+2921215, that further support the measured enhanced variability amplitude in the alkali lines. We provide an artistic recreation of the vertical cloud structure of this AB Pictoris b analog.