As Timothy George points out, Islam affirms that Allah's nature is one (tauhid) in the sense of his utter simplicity and uncompoundedness. But the unfolding truth about God throughout biblical revelation is that God exists eternally in three persons: Father, Son, and Holy Spirit. Islam resolutely denies the incarnation of Christ. Because the deity of Jesus and his place in the Godhead are fundamental to the Christian conception of God, it would be hard to assert, from either the Islamic or Christian perspective, that Muslims and Christians worship the same God.
But when discussing whether Christians and Muslims worship the same God, we must remember that God does not deal with theologies; he deals with persons. Some Muslims are not schooled in theology and approach God with little of this systematic framework. Muslim popular piety is surely channeled by Muslim ritual. But many address God from the heart, with limited theological understanding. While nearly all would deny the Trinity if it were explained to them, their prayers might compare favorably with the experience of Cornelius in Acts 10 and so could be regarded as "God-fearing." Cornelius, of course, was in transit from one religious worldview to another. He was a Gentile whose piety took him to the brink of Judaism and ultimately into the Jesus movement. The majority of Muslims are not that far along.
Still, both in biblical times and in our own, are there not people like Cornelius whose knowledge of God is very limited and yet who turn to him in prayer? We must ask ourselves: How much knowledge about God is required to pray to or to worship God? Initially, not very much. This is not the same as asking, How much do I need to know to be saved? (Even that is surprisingly ...1