In theology two of them claim they do. Muslims claim their god is the god of Abraham, and so do Christians.
How can their radically differing views about God be reconciled though? Christians believe God is triune, which neither Jews or Muslims believe. Similarly, some Jews and Christians recognize Allah's pre-Islamic usage
and argue that Islam's god isn't the same.
Do Jews, Christians, and Muslims actually have the same god all things considered?
Response: Yes. The best way to explain it is that they have the same God, but differ in God's attributes. In Islam, it is taught that all prophets, including Adam, were Muslims. Thus their religion is Islam. For Islam means submission to Allah's will. Since all prophets did this, then by definition, they are all Muslims. Though the name Muslim and Islam was not used. This term was used when Allah sent his last prophet, Muhammad ibn Abdullah.
Yet when Allah sent his messengers, the scripture would end up distorted by the people, resulting in religions that differ, but share similarities as well. This is why Islam accepts Jesus as a Prophet, yet his message has been distorted to the point where he is accepted as God or part of a trinity. Or when Moses was sent to his people to tell them to believe in Allah, and later, the Jews distorted it to where they believe they are chosen and do not accept a prophet after him. So now when Muhammad finally arrives, the Jews reject him as a prophet, and Christians reject him because he does not teach that Jesus is the only way, or that he is God, or that he was crucified.
Simply put, Islam says that they were all prophets sent to teach humankind to worship Allah, which is why Islam accepts them all as prophets of Islam. Whereas, Judaism and Christianity differs. Hence the similarities and differences between each religion, and why they attribute both similarities and differences to the same God.