I've been reading recently about the wars in India that lead to the British East India Company taking over India in the 18th/19th centuries. I've noticed that it's quite common then to both refer each other to the judgement of the Almighty. Of course, this sounds like something either would say to anyone*, but it is peculiar in these situations, given that they believe in a lot of the same things, and iirc, believe in the same God. In practice, I'm wondering if Christians and Muslims viewed each other as sharing something which they did not with other religions (with exception to Judaism)? In more crude terms, was there a sense that the other (Christian to Muslim, Muslim to Christian) was "less of a pagan" than a member of some other religion? That they were worshipping the same God, just "wrong"?
*Edit: At least for "spreading faith by the sword" kind of evangelicals of either faith
The idea that Christians and Muslims "worship the same God" is not really something most people would have believed, historically. Hopefully someone can speak to the British/Indian aspect of your question, but I have some previous answers that might be helpful from the medieval European/Middle Eastern point of view:
I'm a Crusader heading towards the Holy Land in 1096. How much do I understand about Islam?
Why and when did Westerners stop to refer Muslims as Mohammedans?
Basically, Christians thought Islam was a previously-unknown kind of paganism, or some sort of Christian heresy. Or, at worst, it was a deception from Satan and a precursor to the apocalypse.
Muslims also sometimes thought Christians were polytheists. They recognized Jesus and Mary and other Christian figures as prophets of Islam but Christians obviously had an incorrect understanding of the faith, which was perfected by the revelation to Muhammad.
So, if either side recognized that they worshipped "the same God", they simply thought the other side was doing it wrong.