Here's a plot twist, though. Old Testament is canon to Jews, Christians and Muslims alike. The God in there is the same one. It's the treatment of New Testament that differs - Judaism ignores it, Christianity considers it canon and Jesus is God, while Islam considers it a kind of "beta canon", acknowledging Jesus, but only as a prophet.

>But if we both claim to know a guy called John Brown, but I say he's kind, generous, hard working but a little dim and you say John Brown is hedonistic, self-entered and brilliant, are we talking about the same person?

I don't think this example really is catching the core of discussion about Allah (The God). Because we both know a president who bombed two cities and by nuclear bombs. Some people will call him a war hero because he ended WWII, Some people will say he is a war criminal. You will always find something like this when you are talking about a human. Maybe it is also not just different opinions, but different access to information can give us different view on the same person. But we agree that we are talking about the same person, I just find this person to be different from what you think he/she is.

Things are totally different from religious point of view. The discussion between Christians and Muslims is mainly about the nature of The God, not that there is "The God". This also might extend to Judaism, but I don't know much about it.

I agree. And on top of that, neither of us have actually met John Brown face to face. Our _entire_ knowledge of him is based on indirect interactions. We've seen things he's (ostensibly) done, artifacts of his work, we've deduced a set of characteristics about his nature based on his behavior. In a sense, John Brown is nothing more than the label we give to the entity having characteristics A,B,C,D.

So like you say, if the overlap between your A,B,C,D and my A,B,C,D is sufficiently small we should probably acknowledge that it's not the same entity. See also: Ship of Theseus.