I don't know either but I reckon its a lot older than '57. Thinking logically, its a car that poor kids tank around in during the late 50s, so is probably at least 10 years old and I would agree that it looks like a late 40s car. But since when did logic apply in Hollywood musicals?
I think logic and Grease do make uneasy bedfellows. The car does actually fly at the end of the film remember and I don't think Studebaker made a flying car until 1962.