Just to let you know Arabic-speaking Christians refer to God as Allah. I know because I'm Melkite Catholic.
I know. What I meant to say is that the Christian's understanding of God is different from what Muslims have of Allah. The Christian God is not the same as the Allah of Mecca and they were never thought to be alike.
A Muslim will tell you that Allah is not a Trinity so how can it be same deity as the Christian God? You either believe in a Trinity or you don't, but one of the concepts is in error.
Both Christians and Muslims are deluding themselves if they claim that they believe in the same god. Besides, if they really did, Christians would not be persecuted by Muslims as they have been for 13 centuries.
I know this from personal experience; I am a Christian who was born and lived in Muslim countries because my father was in the oil business.
I know that you use "Allah" and the word for "God", but what word do you use for the name of God: "YHWH" or "Jehovah" which is translated in our Bibles into the word "Lord"?