"We Christians joyfully recognize the religious values we have in common with Islam. Today I would like to repeat what I said to young Muslims some years ago in Casablanca: we believe in the same God, the one God, the living God, the God who created the world and brings his creatures to their perfection." - Pope John Paul II, General Audience, May 5th 1999 (emphasis mine).
Can Christians really believe that the Allah of Islam is the same as the Living God of Christianity? Biblically speaking, in a word, no. As early as Deuteronomy 32:16-17, the gods of other religions have been identified as demons. Paul reiterates this in 1 Corinthians 10:20-21. Allah is an idol. He even started life as a literal idol, one of many worshipped by the Sabbeans before Muhammad smashed the rest and declared Allah the only true god. Paul issues a warning to Christians: you cannot drink the cup of the Lord and the cup of demons. So which is it? If you truly worship the same god as Muslims, you are worshiping a demon. Based on the behaviour of Allah, in his purest form (i.e. as he is described in the Qur'an and the Haddith), I would even go as far as to say that Allah is Satan Himself.
As a Christian, I believe things that would get me the death sentence in Islam. I believe that God is triune: Father, Son, and Holy Spirit. I believe that the Father sent the Son to die on the cross so that my sin may be replaced with His righteousness, absolving me of all blame and buying me a place in the Kingdom of Heaven. I believe that through this, Jesus has given me the right to become a son of God, and that it is He who will judge me, and the rest of humanity, on the day of resurrection. In Islam, this is called shirk (as is the Catholic "veneration" of Mary, which I agree is blasphemy, but I thought I'd point it out).
So, do Christians really worship the same god as Muslims? I certainly don't worship a god who can't remember what's written in his own inspired words. The God I worship is far more glorious than anything that can be said of Allah, a demon with whom I refuse to have fellowship.
But there is good news. While Allah is not God, God is also not Allah. Allah is unloving and unlovable, but God loves Muslims. God loves Muslims so much that there is nothing they have done in this life that cannot be covered by the cross. Every Muslim that still has breath in their body has the opportunity to repent of their sin and be granted eternal life. And this isn't a "maybe", as even Muhammad said he didn't know what Allah would do with him (Qur'an 46:9). This is a 100% guarantee (1 John 5:13).