Do Islam and Christianity have different origins?
No. Together with Judaism, they go back to the prophet and patriarch Abraham, and their three prophets are directly descended from his sons--Muhammad (SAW) from the eldest, Ishmael, and Moses and Jesus, peace be upon them, from Isaac. Abraham established the settlement which today is the city of Makkah, and built the Ka'ba towards which all Muslims turn when they pray.
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"When a person dies, his works end, except for three: ongoing charity, knowledge that is benefited from, and a righteous child who prays for him."
Prophet Mohammed (PBUH)
"The best of what a man leaves behind are three: a righteous child who supplicates for him, ongoing charity the reward of which reaches him, and knowledge that is acted upon after him."
Sunan Ibn Mājah
"Every day two angels come down from Heaven and one of them says, 'O Allah! Compensate every person who spends in Your Cause,' and the other (angel) says, 'O Allah! Destroy every miser.'"
Sahih Bukhari