I believe you might have more success with a religious sub, like /r/askreligion or the like. Your question is about current beliefs of current religious groups, and not specifically about the past.

Why do Christians and Jews fight over the same God?

Jews believe in "the" one God. Christians believe in the divine trinity (God, Jesus, Holy Ghost). Considering Jews don't accept that Jesus was even a prophet much less divine, this is an irreconcilable difference.

If Jesus is Jewish, why aren't Christians Jewish?

Christians believe that Jesus was tasked by God (or himself) to reinforce and extend/complete the divine message. I guess you could say they see Judaism as version 1.0, and Christianity as version 2.0. They see no reason to revert to 1.0, when as they see it, Jesus's entire purpose was to give humanity 2.0.

Do Muslims worship the same God?

Muslims worship the same "one" God as Jews. Like Jews, Muslims reject Jesus's divinity, but do believe that he was one of God's prophets (like Moses/Muhammad).