How & why did the (devil/Satan) become the enemy of both the Christians and the Muslims?

Satan in Islam is not the same satan in the bible.

The Bible says that Satan was created by God as a cherub, the most powerful of God's angelic beings.Some time after his creation and before the creation of mankind,5 Satan rebelled against God and took one third of the angels (now referred to as demons) with him into rebellion.

The Satan of Islam was a "Jinn", created from fire as all Jinns were and still are. Angels were and still are created from light. While Angels and Jinns are considered "Angels" in Islam, because they were created from the same material (fire is light, and light is fire), but Jinns are lower than the Angels, because Jinns are like humans; they do sin, while the Angels are perfect and Sinless. The similarity between Angels and Jinns is like the similarity between humans and animals. Created from almost the same material, but two different WORLDS. We also have "human-angels" V.S. evil and bad ones.

Since they are not the same to draw parallels between the two is difficult.