The film is an essay on the pitfalls of conformity. To some degree, especially young people, the need to "fit in" is common enough, and probably has its roots in the desire to be part of a lager group or tribe for survival. Most of us grow away from this tendency. For others, it becomes a lifelong path.
Taken to its logical extreme, however, it can lead someone to a loss of identity, or, as this film later shows, a journey into toxic mass movements like fascism. When economics in a nation are unhinged and troubled, these pressures are naturally greater.
Both this film and director Bernardo Bertolucci's hard-hitting drama "The Conformist" (1970) have similar themes. http://www.imdb.com/title/tt0065571/
Both films draw similar conclusions--that early events in a individual life can lead him into a state of total societal immersion. Both characters turn to the ruthless power of European fascism as a solution to their feelings of personal inadequacy. Bertolucci's character, Marcello, lives out a life of a middle-class father and husband, and also enslavement to Mussolini's spy network. This leads him to a crisis: to become a betrayer of those who trusted him in the past and their murderer, all in the name of scotching dissent against "Il Duce." But does he even care about politics , or is something deeper at work here?
Leonard Zelig chooses total conformity to remake himself from a Jewish/American New Yorker into a storm-trooper in Nazi Germany.
In Allen's movie, there is a ray of hope: a conformist can find a safe haven from extremism through psychiatric care, and, eventually, romance. (This is an American movie comedy after all, and for all of Allen's protests, he follows the rules of the genre. Mia Farrow plays his psychiatrist and just like a good studio director from the era he grow up in, they come together in the end. )
"Zelig" is a lighter comedy for Allen, at least until Hitler enters as a supporting character. But this still is a rare film that is both funny and makes a strong point about how our personalities are shaped by those around us.