Can excessive remembrance of Allah, to the point of repeating supplications a large number of times (such as 2,000 or 5,000 times), cause madness or loss of sanity as is claimed by some people? And was this established in the Prophetic Sunnah?

The remembrance of Allah purifies souls, strengthens minds, and brings tranquility to hearts. It is among the best, purest, and most elevated deeds, and the most safeguarding from Allah's punishment. What you mentioned, that some people went mad because of remembrance, is baseless. You must adhere to remembrance in the manner of the Prophet, peace and blessings be upon him, and his companions. There are many remembrances and supplications detailed in books of remembrance, such as "Al-Kalim Al-Tayyib" by Ibn Taymiyyah and "Al-Adhkar" by An-Nawawi. Our advice to you is to associate with righteous people who adhere to the Sunnah, and to fear Allah in all your affairs.