I think that this issue arises because of a confusion of two aspects of religion. 1. a recognition of what is true 2. the approach to incorporating this truth into our lives.
Monotheistic religions such as Islam and Christianity hold that there is one God. 'The Lord your God, the Lord is One' Deuteronomy 6.4
This also makes sense philosophically, since the omnipotence of God excludes anything that might limit God (so be more powerful). More than one god would limit each other (what one is, the other is not; where one is, the other is not etc...).
If there is only one God, by default, all worship offered to one God must be offered to the same one, even if we argue about who that God is, his revelation and so forth.
It is part of the Christian religion that there is only one God, so there is no possibility that Muslims are worshiping another one, unless of course that god is not God.
From a Christian point of view then, for Muslims to be worshiping God at all, they must be worshiping the only God there is, even if we think they are wrong about who he is and what he has said.
This is the second point. Religions (these religions at least) are not *used* for worship, but are the content of and response to God calling humanity to himself. Yet, this calling has not been universally understood in the same way. So the same God is the object of completely different and competing understandings.
Nevertheless, it might also be worth pointing out that the One God is traditionally called the God of Abraham and Islam considers Abraham to be its father in faith too, although, typically, with quite a different explanation of this truth.
In sum then, Christians and Muslims do worship the same God, but simply because there is only one God to worship. It is just that one of us is worshipping what we don't know (cf. John 4.19-24).
Christianity knows God has revealed himself as Trinity. This revelation is for our good and enables worship to the Father, through the Son, in the Holy Spirit. In this way, God calls us to share in his divine life.