Time-lapse elastic full waveform inversion is used to monitor the spatio-temporal evolution of the CO2 plume during and after supercritical CO2 injection based on a series of time-lapse (repeated) cross-well seismic monitoring datasets obtained at the Nagaoka Carbon Capture and Storage (CCS) site in Japan. The full waveform inversion method successfully estimates the time-lapse velocity decrease of up to 30% within a thin 12 m layer, which is consistent with the magnitude and thickness of the well-log measurements. After the second monitoring survey, the velocity decrease becomes stable and gradually extends down dip along pre-existing geological structures. The full waveform inversion results starkly contrast with the previous estimates based on traveltime tomography. The previous traveltime tomography applications only used the traveltime-delays and resulted in low resolution with few percentage change which was not adequate to correctly resolve CO2 injection changes. The datasets pose significant challenges due to background noise, tube waves, apparent non-isotropic source radiation patterns, apparent reservoir velocity anisotropy and missing key acquisition parameters such as the number of stacks per shot point. To overcome these obstacles, we meticulously perform careful data preprocessing integrating both the body waves and tube waves. We develop waveform-based source mechanism estimation to represent non-isotropic source excitation, and then conduct forward modeling studies to constrain the anisotropy model.