For Christians, the association is metaphorical. You didn't have to be an ethnic Jew to become a Christian, in the early days. On the contrary, the ethos was inclusiveness. However, Christians claim to be worshipping the Jewish God; that their religion represents a new covenant with him.
Similarly, in Islamic tradition the God of Judaism and Christianity is the same Islamic God, although of course the Christians make a mistake by ascribing divinity to Jesus. Mohammad is thought to be descended from Abraham's son Ishmael.
Note that the man is called Abram, not Abraham, at this time. For some reason God changes the names of Abram and Sarai to Abraham and Sarah in Ch. 17.
The Lord had said to Abram, “Go from your country, your people and your father’s household to the land I will show you.In most these tales it is not clear how God is talking to these people. (Later we will see some instances in which there is an identifiable messenger or a communications apparatus. Viz. Moses. For the most part, however, it seems the people are hearing voices in their heads. I must say, the future of the Jewish nation does not seem to fulfill this prophecy very consistently.
2 “I will make you into a great nation,
and I will bless you;
I will make your name great,
and you will be a blessing.
3 I will bless those who bless you,
and whoever curses you I will curse;
and all peoples on earth
will be blessed through you.”
Remember that the Canaanites had been cursed to be slaves. Another curse of God's that didn't land.4 So Abram went, as the Lord had told him; and Lot went with him. Abram was seventy-five years old when he set out from Harran. 5 He took his wife Sarai, his nephew Lot, all the possessions they had accumulated and the people they had acquired in Harran, and they set out for the land of Canaan, and they arrived there.
6 Abram traveled through the land as far as the site of the great tree of Moreh at Shechem. At that time the Canaanites were in the land. 7 The Lord appeared to Abram and said, “To your offspring I will give this land.” So he built an altar there to the Lord, who had appeared to him.I will note that Sarai is 70 years old at this point but we'll let that go. So the deal is that Abraham pimps her out to Pharaoh in exchange for livestock and servants. Quite sporting of her to go along. By the way camels were not domesticated until the 10th Century B.C., about 1,000 years after these events supposedly took place. One more thing -- Sarai is in fact Abram's half sister.
8 From there he went on toward the hills east of Bethel and pitched his tent, with Bethel on the west and Ai on the east. There he built an altar to the Lord and called on the name of the Lord.
9 Then Abram set out and continued toward the Negev.
10 Now there was a famine in the land, and Abram went down to Egypt to live there for a while because the famine was severe. 11 As he was about to enter Egypt, he said to his wife Sarai, “I know what a beautiful woman you are. 12 When the Egyptians see you, they will say, ‘This is his wife.’ Then they will kill me but will let you live. 13 Say you are my sister, so that I will be treated well for your sake and my life will be spared because of you.” 14 When Abram came to Egypt, the Egyptians saw that Sarai was a very beautiful woman. 15 And when Pharaoh’s officials saw her, they praised her to Pharaoh, and she was taken into his palace. 16 He treated Abram well for her sake, and Abram acquired sheep and cattle, male and female donkeys, male and female servants, and camels.
17 But the Lord inflicted serious diseases on Pharaoh and his household because of Abram’s wife Sarai. 18 So Pharaoh summoned Abram. “What have you done to me?” he said. “Why didn’t you tell me she was your wife? 19 Why did you say, ‘She is my sister,’ so that I took her to be my wife? Now then, here is your wife. Take her and go!” 20 Then Pharaoh gave orders about Abram to his men, and they sent him on his way, with his wife and everything he had.Well, you know, Abram lied to Pharaoh and Pharaoh believed him. So why does Pharaoh get the diseases, if this is wrong? Also, if the expectation was that Pharaoh would kill Abram and take Sarai as a sex slave if he thought they were actually married, why doesn't he do it now? And having been taken, why does Pharaoh let Abram keep the loot? Pharaoh, of course, does not worship the God of Abraham, he worships a multiplicity of Gods representing natural forces, social constructs, and abstract concepts. Also, he is himself a God. So there's no reason why he would give a rat's ass what Abraham's God says anyway.
One of the most consistent observations you can make about the Torah is that God is a moral idiot when he isn't being a psychopath. What sort of a moral lesson are we to take from this?