As a muslim let me confirm that Allah is indeed the same God as Christians and Jews worship! We make no distinction about that.
Satan has not tricked mankind into believing that he is God when it comes to Islam (or any other religion). Specifically to Islam this would not have been possible.
The Qur’an was delivered to Mohammad by the angel Gabriel, Mohammad never actually spoke to God directly, Moses is the only one to have ever done that.
Mohammad was able to recognise the devil, one such incident involved the devil revealing that the ‘Ayat-ul-kursi’ (V.2:255) will prevent Satan from approaching you through the night - although the person who he told didn’t realise it was the devil until Mohammad told him.
When Islam was in it’s infancy, many angels fought alongside the Islamic warriors in battle - the devil could not have performed this feat.
Mohammad was actually shown paradise and the universe on one occasion where he was escorted by 2 angels, the devil would not have been able to perform this also.
Everything muslims do is centered around there being one God (Allah), and a firm belief that Satan is bad! If Satan does exist, then so does God, and I don’t think Satan is going to accomplish much by telling us all to worship God and not him!
It’s actually a very interesting question, how do we really know how much power the devil has, and whether or not God would interupt if he (the devil) was misleading us on a mass scale? Anyways, I hope I’ve answered that for you!