Did the Prophet, may Allah bless him and grant him peace, privilege Abu Bakr and Ali, may Allah be pleased with them, with a special dhikr unknown to anyone but them and their descendants who followed them, while disseminating other adhkar to the general Muslim populace?

The scholars did not find that the Prophet, peace and blessings be upon him, singled out Ali or his household with any special knowledge. Imam Bukhari and Imam Muslim narrated that the Messenger of Allah, peace and blessings be upon him, taught Abu Bakr Al-Siddiq a supplication to recite in his prayer. He did not make it exclusive to him or his family; rather, Abu Bakr taught it to the people.

Similarly, Ali, may Allah be pleased with him, denied that the Prophet, peace and blessings be upon him, singled them out with anything, with the exception of what he found in a scroll attached to his sword. This scroll contained curses from Allah upon whoever slaughters for other than Allah, whoever steals land demarcation signs, whoever curses his parents, and whoever gives refuge to an innovator. Ali displayed this scroll to the people and taught them what was in it, which indicates that it was not exclusive to his household.