their defining characteristics of their god preclude the possibility that he and the Christian God are one and the same identity.
“Characteristics” lead to your YOUR objective truth. Not theirs or the Jews.
This is a conclusion you’ve drawn through what you believe to be the objective truth. However their are a Billion muslims who see the Objective truth “different”.
Let alone the fact the the Jews deny that your God is their God. They do not believe in the Trinity, nor that God became man.

While they may acknowledge Jesus was real, so does Islam, in fact Islam believes he was a prophet the Orthodox Jews do not believe this.
To Judaism the Bible states that the Messiah will:
A. Build the Third Temple (Ezekiel 37:26-28).
B. Gather all Jews back to the Land of Israel (Isaiah 43:5-6).
C. Usher in an era of world peace, and end all hatred, oppression, suffering and disease. As it says: “Nation shall not lift up sword against nation, neither shall man learn war anymore.” (Isaiah 2:4)
D. Spread universal knowledge of the God of Israel, which will unite humanity as one. As it says: “God will be King over all the world—on that day, God will be One and His Name will be One” (Zechariah 14:9). But the messiah will not be God.
The historical fact is that Jesus fulfilled none of these messianic prophecies.
Christians counter that Jesus will fulfill these in the Second Coming, but Jewish sources show that the Messiah will fulfill the prophecies outright, and no concept of a second coming exists.
However, we continue to arrive at Christianity, Islam and the Jews and the “attributes” of God which we have already concluded “vary”. Pretty sure we can agree on this?
While your truth is very real to you, their truth is very real to them. So we still must conclude there is only One God. Course less you believe their are other Divine Gods which you would like to inject into this equation? Or are we back to Antichrist or Lucifer came to power? And “if” so does your church state this? Is this your own conclusion? Could you provide the newest document? Just asking.
The Holy Spirit whom Abraham encountered, then Moses encountered, who spoke through the prophets, whom Jesus called “Father” who then through the Archangel Gabriel gave the words of the Holy Qur’an to the Prophet Muhammed …is the same God. This is the claim of all the Abrahamic faith traditions.

Its really that simple. As to who believes exactly what? Not of issue nor which validates one or the other.
All three draw conclusions based on their belief, as to their “own” Objective Truth.
If I place three Chevys in a parking lot, call one Islam, one Christian and one Judaism. All three are still Chevys, though they may be different colors, have different interiors, run different, in fact one might be junk and another a vehicle which runs for 150 Thousand miles, no problem. But they are all still CHEVYS.
So which one is the REAL CHEVY? Yours because you say so?

Or are they all Chevys?
Now as to who is teaching what? Well there may well be some far fetched theory, wouldn’t surprize me. However, we have this in Christinity also? No? Tell me you do not think those playing with Rattlesnakes to see if they are guided by the HS are nuts?
Don’t think we were a little off cue with slavery or was that for “God and Country”? So now we have it all together?
And the arguement I would believe is more like this.
Follow what I’m saying? By your own paradigm your faith is TRUE. Understandable. But do you not see how theirs is also True to them? Thus when you use a Christian belief to weigh Islamic belief, would they not seem like a false teaching to you?
What do you suppose someone of NO-FAITH of Religion would have to say about all of this?
So whats the solution?