Muslims claim they worship the God of Abraham. However if you compare Allah of the Quran and Yahweh of the Torah/Bible, you'l notice Allah is a very different concept. They're not the same. If I brought a unicorn and said this is the god Abraham worshiped, would Muslims start worshiping a unicorn?
Jews don't believe in the Trinity but both Christians and Jews worship the God of Abraham, Isaac and Jacob. Both Christians and Jews believe in the Torah and the other 34 books of the Old Testament as authoritative Scripture. Jesus was a Jewish Messiah. However, He was rejected and crucified. To say Jews rejected Jesus is a false statement. Many Jews DID accept Jesus Christ as the Messiah and Lord. These were the early Christians. But many others rejected Him and continued to live as Jews. In a sense, Christianity is "complete Judaism" in that the prophecies of the Jewish Old Testament were fulfilled in the Christian New Testament.
That doesn't convince me still that Muslims don't worship the same god (of Abraham). To say that we worship the same God is not the same as insisting that we have an agreed and shared understanding of God, which (the latter) is what I think you are portraying, and which I certainly agree.
Christians and Muslims don't worship the same god because we have different understandings of "God". The only similarities between the Christian God and the Muslim god is the claim that they're omnipotent and creator of the universe. There is only one God and therefore only one truth. For example, if I said my god was a pink unicorn from Mars who drank chocolate milk, was all-powerful and was the god of Abraham, would we be worshiping the same god? Of course not! This pink unicorn god does not exist and therefore I would be worshiping a false god. In the same way, God is either Allah or He is Yahweh. He cannot be both. Let me give you a few examples as to why Yahweh and Allah are not the same god. Yahweh's love is unconditional. He loves all people. Allah's love is conditional. He loves only the "believers". Yahweh commanded Abraham to sacrifice Isaac; Allah-Ishmael. Yahweh calls Jews His chosen people while Allah calls them apes (Surah 7:166). Yahweh loves Israel. Allah and Muslims hate Israel. These are only a few examples. Allah and Yahweh are not two names of the same god. One is a false god and the other is the true and living God.
Overnight let me give you an example. Let's say I believed in a god that had six legs, was from Mars, and drank orange juice and I made a claim that my god was the same as your Allah, would you agree with me? Since we have different conceptions of God is exactly why we don't worship the same god. If you claim Christians and Muslims worship the same god, I'll assume you worship Jesus becuase in our eyes, He is the God of Abrahm, Isaac and Jacob.