It’s becoming really sad watching you dance around this issue. It doesn’t matter what the Muslims say. It’s obvious on many levels that they do not serve the God of Abraham, Isaac, and Jacob. That the Catholic Church doesn’t acknowledge that fact is evidence of their not even understanding the difference.
I can see by your language, including changing to use the word serve in place of "worship" that you have lost the debate regarding whether Moslems worship and adore the God of Abraham, although they do so in serious error. They acknowledge Him as "The God" by the same name you wrote. This is unrefuted.
How far does your exclusivity carry ? Do you agree that the Jews who categorically reject the books of the New Testament, Jesus as Messiah, and Messiah as God, worship and adore the God of Abraham, Isaac, and Jacob ? They certainly assert that they do ! Do you maintain they serve a different God of Abraham, Isaac, and Jacob than you ? Do Catholics worship and adore the God of Abraham ? Do Protestant faith communities, sects, and cults worship and adore the God of Abraham ?