Why do Jews Christians and Muslims believe that Jerusalem is their holy city?

Answer 1

None of these religions purports that it is exclusively "their holy city", i.e. its mine and no-one else's, although the Islamic Waqf of Al-Aqsa Mosque and the Haram Al-Sharif routinely tries to discredit the Jewish connection of the city in order to buttress Palestinian claims. (The Waqf somehow avoids the problem of Christianity holding the city as holy, though.)

Jews acknowledge Christian and Muslim reverence of the city and this is most clearly noted by the permission given by Israel for Muslims and Christians to visit and regulate their holy places in the Old City.

Christians acknowledge the Jewish history of Jerusalem since it undergirds Jesus' activities in the Holy Land. Christians do not actively recognize the Muslim holiness of the place, but have not attempted to refute it either.

Proper Muslims acknowledge both the Jewish and Christian reverence of the city since King David, King Solomon, and Jesus are all Islamic Prophets.

To find out what particular parts of Jerusalem each finds holy, please see the Related Question.