It's not even remotely the same. Not even close. Islam and Christianity have some fundamentally different, incompatible beliefs about the nature of God. The Quran explicitly condemns (an apparent misunderstanding of) Christian orthodox teaches on Trinity and incarnation, which are absolutely central to Christianity but anathema in Islam. The Christian community, likewise, tended to see the early Muslim community not as a separate religion but as Christian heretics, that is, as a community defined by its egregiously erroneous teachings on the nature of Christ and his relation to God.

They certainly are not the same religions. Islam originated, in part, as an explicit rejection of the Christianity that surrounded it, because of these fundamental disagreements over the divine.