I know they used to be called the “People of the Book,” but were they aware of the shared ancestry of their religions, so to speak? Or was there a time when Christians or Muslims or Jews did not see it that way? Thank you!
That really depends on how you define the term "same God" on the most basic level yes. Christians knew that they were "descended" so to say from Jews and Muslims from Christians.
Where your question becomes a bit less clear is as I have stated the term "same God" because of the Christian belief in the Holy Trinity something that at times has been argued is polytheistic. So not to go into theological debate yes they have always known that they are all Abrahamic, but their disagreements could be defined as saying they worshiped a different god.