Since this was posed as a "theology question," not a "church-history" or "comparative religions" question, I think the proper answer is that Christianity, Judaism, Islam, all the various competing sects within each tradition, and virtually all other contemporary religions worship the same God, insofar as they all agree that only one God exists.

That might be the only thing they all agree upon, of course, and each faction believes that it has the best concept of exactly Who that God IS, and how we humans are supposed to relate to Him..

Monotheism has been pretty well established among believers of all stripes for a very long time. "God" is essentially a name for "the Truth," the nature and ruling principle of the entire universe, seen and unseen, well beyond human understanding. And everyone can agree that there's only one ultimate truth, unknowable as that might be and however much we might disagree about it's nature.

Even Hindus, who honor and practice a very ancient "polytheistic" tradition, will describe their multiple deities or demigods as expressons of the various aspects or personalities of One God, much like the Trinity posited by Christians, or the many saints of Catholicism, the multiple demigods of voodoo and santeria, etc.

Once upon a time, when one "people" or tribe went to war against another, they actually believed that their own "god" would be facing off in opposition to the enemy's "god." That concept is long gone. The idea of one God common to all men and all creation goes back at least as far as Abraham. If one group of his worshippers thought of themselves as a "chosen people," they were claiming to be the only ones who knew and faithfully tried to follow the creator of all ~ they were quite explicitly NOT following the notion that "we have a god who is not your god."