I do not consider the Abrahamic God to be a false god. Differences in dogma and concepts of worship do not change the reality that Muslims worship the same God as you or I do. Maybe God gives a damn about such differences, maybe He doesn't.
It depends on the meaning of the word "worship."
If "worship" to a muslim is killing infidels in the name of their God, is that the same God as Jews and Christians worship.
It seems to me the muslims have both the dogma and the God wrong.
The muslims' founding prophet was clearly a man of violence, so it would follow their beliefs are not Christ-like. Hence the wrong God.
The muslims' God is the God of Muhamed., not of Christ.