So, who is the god of Abraham? Is it Yahweh, God, or Allah? Or, are they all simply one in the same?
It depends on how you define things. The Christian god is supposed to be the same god as Allah, but obviously you can’t actively believe they are all one in the same. Why? Because of the differences of laws, beliefs, and the judgments those gods are involved in. A Christian is not going to look at the things attributed to Allah, such as the prophet Muhammad and the insistence that Jesus was only a prophet himself, and say that it fits in line with their beliefs. It just not going to happen.
“Yahweh” is the original incarnation of this deity, so far as we know, with the Christian god and “Allah” as basically evolutions: The Christian deity requires the Jewish deity to become a different god, just as the Islamic god requires certain Christian influences – all while remaining “separate-but-equal” deities.
So, while you can easily claim they are based on the same original deity, it’s actually quite confusing to insist them to be the same deity now – or the trinity concept is meant to refer to a rather interesting case of a god having a multiple personality disorder. While busy on this subject, I ought to start a new holy book based around the next evolution of the same deity. We’d also get to see first-hand how that same supposed Poly/Mono-theistic duality of this god evolves.
One of the personal issues I have with the deities of religions that are “borrowed” from other religions is that it often points to the human influence. Gods which gain or lose specific traits, in regards to their previous versions, seem to change over time and end up just as variable as humans. We change due to our influences, and apparently so do these supposedly all-knowing gods.