Do Muslims and Christians worship the same God, only by a different name?
Clearly, when Muslims and Christians (and Jews, Hindus and others) speak of God, they are all referring to the uncreated Creator of the universe. Moroever, Islam embraces monotheism, and draws its teaching from many of the biblical characters (including Noah, Jacob, and Jesus).
But the fact that the God of Islam has something in common with the Christian God does not mean that they are identical.
As Ergun Mehmet Caner explains (in the Apologetics Study Bible):-
The God of Islam is remote, and not intimate (Surah 112). Allah does not have a son (Surah 2:116), is not a Trinity (Surah 5:72), and does not love unconditionally (Surah 8:53). In Islam, Jesus is simply a prophet, and emphatically is not the Son of God (Surah 5:72), and the angel Gabriel is the Holy Spirit (Surah 2:193).
In other words, ‘Islam rejects the fatherheart of God, the divinity of the Son, and the person of the Holy Spirit.’
Chalk and cheese may have a number of things in common; but that does not make them different words for the same thing.