What is the Islamic concept of Bai'ah? How is it given? I recently went to a lecture where they were talking about repentance. They asked those present to give Bai'ah to the scholar who was giving the lecture, then asked those present to ask Allah to repent from their sins. They said this is the way of the Sunnah.

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Bai'ah (pledge of allegiance) in Islam is the pledge given to the ruler of a region who is accepted by its people.
Ibn Khaldun defined Bai'ah: saying: 'It is a pledge of obedience. It is as if the person who gives the pledge of obedience to his ruler promises that he will accept his rules concerning his matters and the matters of all Muslims as well. He will never dispute with him and will follow his orders willingly or unwillingly. The people used to put their hands in the hand of the ruler when they gave the pledge of allegiance to confirm the pledge just as a seller and a buyer do to confirm their contract. For this reason, generally the Bai'ah was associated with shaking hands' .
What is mentioned above is the concept of Bai'ah in Islam. Anyone who explains the Ahadith of Bai'ah contrary to the above meaning is an innovator and goes against the practice of Ahl-al-Sunnah. Therefore, what is mentioned in the question is not proved in Shariah and not reported from the good ancestors that they used to take Bai'ah for repentance. They only made Bai'ah with the ruler of the people.
Allah knows best.