It used to be said the Christians and Jews worship the same God. Now we have the Muslims added into the soup, with the Islam-favoring category of the "Abrahamic" religions.
Do Jews, Christians and Muslims worship the same God, only in different ways?
Jews worship a monotheist God with a unique relationship to Israel. Christians worship a monotheist God who is also a threesome of Father, Son and Spirit and the Son became human in Jesus. Muslims worship a monotheist God who is most definitely not any kind of threesome and whose revelation culminates not in Moses or Jesus but in Muhammad.
The younger religion has a stake in making a case for "the same God" because it legitimates its claim against its ancestor. It can claim what it likes in the ancestor religion and still maintain its supercessional supremacy. But the ancestor religion is far less likely to recognize its God in the version maintained by the new kid on the block. Would a Jew recognize as "his" God the Trinitarian Christian God? And would a Christian recognize as "his" God, the anti-incarnational Unitarian Muslim God?
Bahai is a post-Islamic religion which tries to do to Muhammad what Muhammad tried with his predecessors, include him in a larger vision in which he loses the place he asserted for himself and is relegated to an ancillary role, preparatory to Bahaullah. Would Muslims agreed that they and Bahai's worship "the same God"?
So I am inclined to say that "we" monotheists of the Adamic religions do not worship the same God.
"We" compete with and against each other for ownership of the God who created Adam and Eve. It's sort of like the Palestinians and the Israelis saying, "We both love the same land".
*The photo shows demonstrators in Pakistan supporting the death sentence given to a Christian laborer accused by her Muslim co-workers of blashpheming Muhammad.