Christ is a title - it is the Greek translation for the Jewish term "Mosiach" which means the annointed of God.
"Jesus" birthday was assigned to December 25 because it was originally a pagan holiday marking the "Rebirth of the Sun". When Constantine got the Christian Bishops together at the council of Nicea they were mandated with the creation of a religion that would unify a failing Roman Empire. Pagan holidays were given Christian context- so the "Birth of the Son" was assigned to the day that was previously assigned to the "Rebirth of the Sun" after the winter solstice.
Note - Jesus is a Greek name. If there was a real person that Jesus was based on his name was probably "Yeshua".