I can deny my parentage. Hell, I'd like to. That doesn't change my source. As I said previously the three fork from the same original stories. They differ in interpretation, not source.
That's taking it as a matter of history and mythology, first of all.
But, even then, consider this - we are not discussing whether they all come from the same root. We are discussing if the gods are the *same entity* in some sense.
Using your own analogy - you are not the same individual as your mother, are you? No, you are not. You are a mixture of your mother, and your father, and your upbringing, that lead to a completely separate individual - you may share some similarities to your ancestors, but you are not the same person. So, apply that to the Abramamic god(s). The Judaic god gets reformed into the Christian god. As myth and literature, Islam borrows from Judaism directly, and from Christianity, and from Persian and Arabic mythology, and changes it all and wraps it up together to get Allah.
Now, at any of these steps, as mythology and literature go, we might call the god the same entity, or we might say that the changes at one or more of those steps yields an entity that is sufficiently different from what came before as to not be the same thing.
As a pretty direct analogy, we will take the Norse god Odin.
The Odin most of us know is a war god, king of the Norse pantheon, called "All-father". He's the Big Cheese of the pantheon. But, note that the Norse didn't always have him that way. The war-god Odin is the god of the late, Viking-Norse. To the earlier Norse, before the Viking period, however, his brother Tyr was the king of the pantheon - and Tyr was more a god of justice than war (not strange, for a people who had not yet turned to aggression and raiding as a notable economic pillar). Odin had a role in the stories more equivalent to Hermes or Mercury - the messenger deity, rather than the king and war god.
If we told a story of the old Odin, without naming him, you probably wouldn't recognize him. The Odin we are most familiar with has the same name, and is said to have some of the same history, but some events previously ascribed to Tyr and others then gets ascribed to Odin, and he was transformed enough that he even takes a different role in the mythology - it would be reasonable to say that the older version really was a different entity by the same name.
And all that is setting aside the more philosophical and theological question - say one or more of the gods of Judaism, Christianity, and Islam *actually exist*. Are they actually all worshiping the same being?