Obviously individual Christians and Muslims will say that, but as a whole, Islam considers Christianity to be a “religion of the book”.
The actual term is “People of the Book” (Ahl al-Kitab), which refers to those religious communities (Christians, Jews, and I think also a few others who aren’t as well known like the Manicheans and Sabians) who Muslims believe received some sort of divine revelation in scriptures prior to the coming of Islam. It doesn’t really say anything about whether or not they believe a particular group to be worshiping the right God, as you may or may not know Muslims argue that Christians and others distorted their scriptures (with spurious, ambiguous references from their Qur’an to back up their view).
Islam, like Christianity, tended to obliterate Pagan temples, but would convert churches into mosques because they were already houses of God (not exactly a ringing endorsement of Islam here…

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I take it you mean “…but would NOT convert churches in mosques”? Otherwise that sentence doesn’t make sense. It’s wrong, anyway. Some of the most famous mosques in the world (e.g., the Umyyad mosque in Syria) used to be churches. In some places in the world (e.g., Turkey), the majority of mosques dating before the modern period were originally churches. In Spain, buildings that were originally churches changed back and forth with the shifting forces of the communities. This is why many of the older churches in Spain have a certain “Islamic” look to them (which is really the Islamic recension of Eastern Christian architecture, stripped of its obvious Christian symbolism).
But one Caliphate in medieval Jerusalem chose not to enter the Church of the Nativity to worship specifically out of respect for Christianity’s holiest site - if he worshiped there, he feared Muslims of later generations would seize it and worship there.
Well that was awfully nice. It’s a shame that there is no consistency in this regard. In subsequent centuries (and especially in the modern era) there has been no respect for anything Christian in the holy land, as political conflicts have sacrificed our villages and holy sites, either at the hands of the Muslims (e.g., the shootout in which Islamists took up positions inside the church a few years ago, leading to much damage), or Jews (e.g., starving via blockade and media blackout the
peaceful protestors of the Christian-majority village of Beit Sahour during the first intifada).
Islam, as s whole, believes Christians and also Jews worship the same God.
This is an Islamic talking point, meant to open the naive to their evangelization efforts and nothing more. Don’t buy it. They mock us for worshiping a “God who eats”, or a “God who dies” or whatever.
They just believe that they have the fullness of Divine revelation, believing Mohamed to be the final prophet, rather than Jesus, whom they consider a powerful prophet who rose from the dead.
They don’t believe in the resurrection, because according to the Qur’an Jesus never died on the cross. It’s kind of hard to come back from the dead without dying in the first place.
Edit: Yeah! What Jakasaki said!

