There were no Moslems prior to 632 in the Middle East. By about 732, there were very few Christians or Jews, because they had been slaughtered or forcibly converted. The Moslems invaded all the way across North Africa, through Spain, and well into France before being stopped. Islam was spread by force of arms.
My understanding is that it we have now goes away back to the birth of Ishmael.
Genesis 16:11-12
11 And the Angel of the Lord said to her:
“Behold, you are with child,
And you shall bear a son.
You shall call his name Ishmael,
Because the Lord has heard your affliction.
12 He shall be a wild man;
His hand shall be against every man,
And every man's hand against him.
And he shall dwell in the presence of all his brethren.”
Genesis 17:20-21
20 And as for Ishmael, I have heard you. Behold, I have blessed him, and will make him fruitful, and will multiply him exceedingly. He shall beget twelve princes, and I will make him a great nation. 21 But My covenant I will establish with Isaac, whom Sarah shall bear to you at this set time next year.”
So is what we know as Islam today, are they the lineage of what we see today?
If so his descendent's have a right to claim to be descendent's of Abraham.
Jesus said
John 16:2-4
2 They will put you out of the synagogues; yes, the time is coming that whoever kills you will think that he offers God service. 3 And these things they will do to you because they have not known the Father nor Me. 4 But these things I have told you, that when the time comes, you may remember that I told you of them.
“And these things I did not say to you at the beginning, because I was with you.
Whilst Jesus was talking about the imminent persecution it also relevant for today.
WHOEVER KILLS YOU THINKS THAT HE OFFERS GOD SERVICE.
Jesus said
John 8:37-44
Abraham's Seed and Satan's
37 “I know that you are Abraham's descendants, but you seek to kill Me, because My word has no place in you.
38 I speak what I have seen with My Father, and you do what you have seen with your father.”
39 They answered and said to Him, “Abraham is our father.”
Jesus said to them, “If you were Abraham's children, you would do the works of Abraham.
40 But now you seek to kill Me, a Man who has told you the truth which I heard from God. Abraham did not do this.
41 You do the deeds of your father.”
Then they said to Him, “We were not born of fornication; we have one Father—God.”
42 Jesus said to them, “If God were your Father, you would love Me, for I proceeded forth and came from God; nor have I come of Myself, but He sent Me.
43 Why do you not understand My speech? Because you are not able to listen to My word.
44 You are of your father the devil, and the desires of your father you want to do. He was a murderer from the beginning, and does not stand in the truth, because there is no truth in him. When he speaks a lie, he speaks from his own resources, for he is a liar and the father of it.
That also applies to Islam as we know it.
So therefore it is possible they believe in the same God as us but they hold the teachings of allah over that of God.
Allah is off the devil hence so is Islam.
My thoughts