One objection to the claim that Muslims and Christians worship or believe in the same God is the statement that the Christianity Deity is the Trinity, and that Christians worship Jesus as God–contrary to Muslims.
Without revisiting the issue of linguistic reference, I'd just point out that this understates the difference between Christian theism and Muslim theism. This approach defines the difference in terms of what God is or the attributes of God. God as Trinity. God as Incarnate.
And that's certainly valid as far as it goes. But there's more to it than that. In the Bible, there's a sense in which God is what God does. By that I mean, God's actions often mirror his nature or character. Actions correspond to attributes.
To reject the Incarnation is to reject what God does as well as what God is. Why do Muslims repudiate the Incarnation? It's not like dying that George Washington cut down a cherry tree. Even if that's an apocryphal story, it's possible that he did that. For Muslims, it's not just a case that God didn't become incarnate, but that he couldn't become incarnate. Allah is not that kind of God. To Muslims, it wouldn't be fitting for God to become incarnate. That's an inherently improper thing for God to do.
Likewise, Islam as a doctrine of revelation rather than inspiration. That's why Muslims are confused when Christians say the Gospels are the word of God. For Muslims, that's contradictory inasmuch as the Gospels were written by men. They dichotomize divine and human action. God can speak to people, but he can't speak through people. They don't have that framework. Human action picks up where divine action leaves off, or vice versa. If God did it, man didn't, and if man did it, God didn't. Allah does not and cannot interact with humans.
By the same token, Islam is a religion of revelation and duties rather than a religion of redemption. A religion of prophets rather than redeemers.
God sends prophets to inform humans of their social and spiritual duties. Allah remits sin apart from atonement. Islam is voluntaristic.
Allah isn't the kind of God with whom you can be in a right or wrong relationship. In many ways, Allah is more like the ground of being. You can't have a right or wrong relationship with the ground of being. It's simply something you depend on. Something that makes other things possible.
In Christianity, by contrast, for sin to be forgiven, sin must be atoned. That's why Christianity, unlike Islam, has a theology of vicarious atonement and penal substitution.
It involves a different kind of God. Justice is a divine attribute. Injustice must be satisfied. In addition, sin changes the relationship between God and man. Conversely, justification changes the relationship between God and man.