Is the Mormon Jesus the same as the Jesus of the Bible?

No - not at the level that matters most, the doctrine of God. The name is shared, but Scripture's Jesus and the Jesus of Latter-day Saint teaching are set within genuinely different accounts of who God is. In its own teaching, the LDS Church holds that the Father, Son, and Holy Ghost are three separate beings, with the Father and the Son having bodies of flesh and bones (Doctrine and Covenants 130:22). Its most distinctive teaching, summarized in Lorenzo Snow's couplet, is that God the Father was once a man who progressed to godhood.

The historic Christian faith, shared by Catholics, Orthodox, and Protestants, confesses something different: one God, numerically one divine being, who exists eternally as three coequal, coeternal persons. He did not become God, and there was never a God before him. Isaiah's monotheism is sweeping here: before me no god was formed, nor shall there be any after me. So the disagreement is narrow and sharp: is God one being or several, and did God become God, or is he eternally and unchangeably God?

The honest conclusion is that the two share a name but describe genuinely different beings - a different account of God, and so a different Jesus. That is not a reason for contempt or for doubting anyone's sincerity; it is a reason for the kind of clear, gentle conversation that takes the difference seriously precisely because the answer matters so much.