When did Muslims begin to see their religion as a new faith that was meaningfully distinct from Judaism, Christianity, and Zoroastrianism?

7th century CE through Prophet Muhammad.

Explanation:

Muslims began to see their religion as a new faith during the 7th century. Prophet Muhammad established these new faith in an attempt to unify the Arabic world under Islam.

Prophet Muhammad envisioned that God came to him and inspired him the write the Qu'ran, the Islamic religious book. It contains all the religious tenets of Islamic religion.

Though Islam originated in the Middle-East, it is also meaningfully distinct from other Middle-Eastern religions in many ways. Unlike other religions, Islam has certain strict conducts of life and set of rules that every man and woman should follow. It's religious texts also contains many instructions of conducting daily lives.