Is the communal recitation of the Hizb after the Fajr and Maghrib prayers authentically established from the Messenger of Allah (peace and blessings be upon him)? And what is the ruling on communal recitation with the intention of memorization?

Reading the Hizb collectively after Fajr and Maghrib prayers was not established during the time of the Prophet (peace be upon him) nor during the blessed generations. Therefore, consistently performing this reading specifically after Fajr and Asr, while believing it to be a Sunnah, falls under additive innovations (Bid'ah Idhafiyyah). It should thus be abandoned. One can memorize the Qur'an without falling into innovation.