Recently I had encountered a couple of exchanges where Islam and Sufism were claimed to have influenced Vaishnavism (Vedic Brahmanism in its new avatar) and Advaita respectively. This is my customary abrasive rejoinder.
To begin with there is little doubt that the Philosophies in Islam and Vaishnavism are the same. But that should hardly be a surprise. Almost all the proletyzing religions of the world (Zorastranism, Vaishnavism, Judaism, Christianity, Islam) have the same philosophy. For afterall they all have the same central asian origin and have had extensive exchange of ideas thereafter.
Proletysers like Zakir Naik have preached ad-nauseum, the similarity between Vaishnavism (which he mistakenly calls Hinduism) and Islam. Stuff like Allah-Brahman (one without the second, transcendant God), 99 names - 108,1008 names (attributes), Angels-Devas (lesser deities), Muhammad-Krishna (prophets), People of the Books (Torah, Gospels) - Vedic People (Vedas, Upanishads), Quran-Gita (latest message), Jihad-Arjuna (war as karma) are too hard to miss. Ofcourse this similarity is used to encourage more conversions to Islam.
Infact Naik goes on to claim that the so-called Bhavishya Purana had predicted the birth of Muhammad and that Kalki avatar is none other than Muhammad. Such claims, however interesting, are juvenile and should not be taken seriously. All the Abrhamic religions (which I now think also includes Vaishnavism!!) come from the same lie that there is a monotheist creationist God who will judge us. Borrowing Einsteins infamous words, once we understand the bakwaas at the root, we can work out the details.
But claims that Islam/Sufism influenced Vaishnavism/Advaita are clearly misplaced and mischievous. As discussed above similarities between the practices of the Abrahamic religions are well known and documented. Dara Shikoh, the greatest Mughal Emperor we never had, went so far as to say that the Kitab-al-maknun (the hidden book) mentioned by Prophet Muhammad in the Quran was the Upanishad. It is preposterous to rearrange the pages of history to claim legitimacy for one's religion.
As for Sufism influencing Sankara's Advaita, its sheer non-sense. The confluence of the Transcendant and the Immanant God had been made as early as the Upanishads/Gita. Islam, Sufism and Vaishnavism are all Monotheist, not Monist. Advaita on the otherhand was a monist reading of the Upanishads which themselves were heavily influenced by the Shramanic and mostly atheist schools of the non-Brahmin scholars of Magadha (Ajivikas, Carvakas, Jains, Buddhists). Sufism resembling the Vaishnavaite Bhakti movement of Andal, Meera, Chaitanya etc is common-sense. People like personal access to God.
Muslims should stop being paranoid and begin to investigate Muhammad more honestly. Christians are no longer shy of accepting that Jesus got no divine message from any God but produced his own doctrine from his travels. There is even a movie going to be made claiming that Jesus visited India to learn/teach from scholars there. (Missionaries can rejoice!)
Consider few ideas. Muhammad was a merchant traveler and might have exchanged opinions with traders from Persia, Central Asia and South India. Vaishnavism took formal monothiest and sufi like devotional shape in South India roughly before the time of Muhammad. European Spice trade with the Indian Malabar coast was carried through Egypt and Arabia. Why could not Gabriel, the 'angel' who taught Muhammad be a Mallu from Kerala then?