Générique relates to genre and specially engendrer and lists all the people who generated (gave birth to) a movie, its géniteurs or générateurs. The word is attested in the early fifties and started to be popular in the sixties.
As Circeus commented, using générique for a theme song came later by metonymy.
Google Ngram:
Source: Émile Moussat, Ce que parler veut dire, Sixième série, p 134, 1960.