Technically the Jew's God and the Christian's God are the same, but the Jews don't recognize Jesus as part of that essence.
Allah and the Christian God are NOT the same. The Christian God is a trinity and He is a personal God.
- In the Qur'an, Sarah 19:92: "For it is not consonant with the majesty of ((Allah)) Most Gracious that He should beget a son."
- In the Bible, John 3.16: "For God loved the world so much that He gave His only-begotten Son, that whoever believes in Him should not perish but rather have eternal life."
- In the Qur'an, Allah is identified as "Allah the loving" and he shows his love to people, and yet Allah doesn't love unbelievers or the ungodly.
- The Christian God loves the whole world (Jn. 3.16). God loves us even while we are sinners (Rom. 5.8).
- Allah is not personal. In Islam, one submits to God and shows his/her devotion by following the five pillars of the faith. The Christian God is personal. He speaks and He seeks us to draw us to Himself.
Also, God who has revealed Himself in the Bible and god who has revealed himself in the Qur'an are completely different revelations. The two "holy" books contradict each other, so it's impossible that both are true or that both are the revelation of the One True God. Therefore one of the books is untrue, and therefore the god it points to is not the true God.
I have a chart on my computer that compares and contrasts what the Qur’an says Allah is like with what the Bible says Yahweh is like. They are as different as night and day. Allah and Yahweh can only be the same God if we accept that both religions claim there is only one God, but we reject what the Qur’an says and believe what the Bible says, as the Muslim-background believers say.