Novel back story
Jason Bourne is but one of many aliases used by David Webb, a career Force Reconnaissance Marine Captain Foreign Service Officer and a specialist in Far Eastern affairs. Before the events in The Bourne Identity, Webb had a Thai wife named Dao and two children named Joshua and Alyssa in Phnom Penh. Webb's wife and two children were inadvertently killed during the Vietnam War when a fighter plane strayed into Cambodia, dropped two bombs, and strafed a spot near the Mekong River. However, unknown to Bourne, Joshua survived.[4] Due to Cambodia's neutrality in the war, every nation disclaimed the plane, and, therefore, no one took responsibility for the incident. Infuriated by both the utter injustice and randomness of his loss, Webb went to Saigon and, under the careful guidance of friend and CIA officer Alexander Conklin, ended up training for an elite Top Secret Special Forces unit called Medusa. Within that select organization Webb was known only by his code name, Delta One.
Under Operation Treadstone section: At this point, Webb (Delta) takes the identity of Jason Bourne due to the actual Bourne's status as MIA in the war as well as the fact that Bourne was in reality a ruthless killer with a long criminal record. The point of all this was to turn Jason Bourne into something more than he really was, a contract assassin who would be known all over the world for terminating the lives of just about anyone.
Film version
The ploy worked; Webb approached the CIA and was brought into the program by Neil Daniels, a supervisor in the Treadstone program, and was monitored by Dr. Albert Hirsch, who supervised the medical assessment of the Treadstone agents. He was brought into a secret recruitment center in New York City, where Hirsch ordered him tortured for days — via waterboarding and sleep deprivation — to break his spirit and allow him to be molded into an assassin. He was officially accepted when he murdered an unidentified man (implied to be the real Jason Bourne) without questioning. After that, he was given a new identity as Jason Bourne, and his birth identity was dropped.
His real identity as David Webb was scrubbed, and he was renamed as Jason Bourne