A brief answer is "No, they do not." The reason is simple: Christians worship a trinune God. Muslims do not. Christianity affirms that there is no salvation outside of Christ. Muslims, while having much good to say about Jesus, deny that Jesus died on the cross and thus nullify everything that the New Testament affirm about Him and what it means to worship God.
Some have argued that, while the answer above may be said to be theologically correct, it misses nuances in the question. It has been suggested, e.g., that they do worship the same God but the Muslim—by failing to know Him (or refusing to accept Him) as the trinune God—is worshipping Him defectively, perhaps even rebelliously. This would also be true but then the question has shifted significantly, and not just in nuances. At Sinai the Israelites got Aaron to make them an idol in the form of a calf and worshipped it. In their worship of the calf, Aaron affirmed that their's was "a festival to Yahweh" (Exo 32:5). But it mattered little what they called it, it was idolatry, and as far as the Lord was concerned, Israel had "become corrupt" (v7). And that seems to be the point of the Second Commandment, that Yahweh is not worshipped when He is worshipped defectively. The name in which such defective worship is offered carries no import: if such worship is repugnant to the God who is claimed to be worshipped, that God is not worshipped.
Resources:
R. Albert Mohler, "Do Christians and Muslims Worship the Same God?" Decision 1 February, 2016.
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Duncan Peters, "The 'Same God' Issue and the Communication of the Gospel to Muslims," Foundations 82 (May 2022):23-32.
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