We’re all floundering here because of our misconceptions about what “same god” means.
First of all, does it make any sense to ask if they’re praying to the “same god” if, for instance, there were no gods? What would the statement “we pray to god X” refer to if “god X” doesn’t actually exist? Surely we mean to discuss this in a way that allows for that possibility, and so we don’t really mean “same god” the way we might mean “same person” when discussing the discovery that we had a mutual friend.
I think what we mean when we talk about the “same god” is what we mean when we say figuratively about a person who has changed that “he seems like a different person.” If I knew person X when he was young and rowdy, and you knew person X when he was middle-aged, forlorn and regretful, you might say “are we talking about the same person” but you would not mean it in terms of actual identity. What we’re talking about would be the person as defined by his characteristics, rather than by identity-over-time. With the person we have the luxury of knowing he exists and can make the identity-over-time observations and even test his DNA to establish he’s the “same person”, but with “god” we don’t know that (at least not in the same way) so we have to identify “god” by attributes instead of by singluar identity.
If we identify a “god” by its attributes, then the Christian god and the god of Islam are certainly different gods. They have completely different natures, have done different things (which also reveals character differences), they request different things of their followers. If a god’s identity is established by the set of its character traits, that god is his attributes, such that god X = {X1, X2, X3, X4,…} and god Z = {Z1, Z2, Z3, Z4, …}, then X = Z if and only if all the elements of the sets are identical. They are clearly not identical between the god described by Christians and that described by Islam, so they are not the same god.
There are a couple senses in which you could argue they are. The religious heredity argument that seems to be predominating the discussion is one of these, both religions sprang from a common earlier religion. Really, this only means that as a consequence both have given their god the same “name”, the same initial referent statement identifying him as the god who appeared to Moses at a certain time under certain circumstances.
In a tantalizingly logical sense, there is a way in which all monotheistic religions honor the same god. Since the primary principle of monotheism is that there is one and only one god that actually exists, then any two people both affirming a belief in monotheism are talking about an entity that shares at least one important characteristic, that of being the only god. If there really were a god, and so the statement had an actual referent, then they would have to be identifying the same thing at least in the sense that identity is based entirely on such a fundamental attribute. By analogy, I’m “me” because everyone else isn’t. So god is “god” because nothing else is. In this sense, they would be naming identical entities, but of all the monotheist traditions in the world and all that might be hypothetically posited, all but one of them would of necessity be monstrously ill-informed about the actual attributes of that entity.