Is it correct to say that the God of Christians and the God of Jews is the same God worshipped by Muslims, and what is the authenticity of this assertion, and are there any caveats to it?

The statement that the God of the Jews and Christians is the God worshipped by Muslims is correct. Muslims and the People of the Book worship one God, who is Allah. Quranic verses have affirmed this. However, the People of the Book associated partners with Him and attributed imperfections to Him (such as fatigue, stinginess, and regret), and disbelieved in some of His messengers, such as the claim that Uzair is the son of God (Jews), or that the Messiah is the son of God or one of a trinity (Christians). Meanwhile, Muslims are devoted to Allah alone and associate nothing with Him. This polytheism (shirk) and disbelief (kufr) rendered their original faith in God of no benefit to them.