The fast radio bursts (FRBs) are energetic radio bursts with millisecond duration only observed at radio frequencies. The generation mechanism is still mysterious. We have proposed a generation mechanism of both repeating and one-off FRBs. They arise from axion star collision with a neutron star or a magnetized accretion disk of a galactic black hole. Once we accept the existence of the axions, we find that the mechanism well explained previously observed spectral-temporal features. In this paper we show that it also explains recently observed phenomena such as downward drifting in the repeating FRBs, etc. Analysis of the downward drifting based on Doppler effects has been presented in recent papers, in which a superradiance system of molecular or atom has been proposed as a source of FRBs. We apply the analysis to our mechanism and find that it well explains the relation between the downward drifting rate and the duration of the repeating FRBs. The Doppler effects lead to the fact that the duration of radio burst with higher center frequency is shorter than that of radio burst with lower center frequency in the repeating FRBs. Our generation mechanism naturally explains polarization angle swing observed in the repeating FRB180301 and one-off FRBs. We also discuss the association between the FRB200428 and magnetar SGR J 1935 +2154 . The x-ray burst observed just after the observation of the FRB could be triggered by the axion star collision with the magnetar. We also explain the consistency of our generation mechanism with observed spectral-temporal differences in the repeating and one-off FRBs, e.g., longer duration (smaller luminosity) of repeating FRBs than duration (luminosity) of one-off FRBs.