Based on the front of the pass (notably the payment card symbols, the AT&T symbol and driver's licence type notification), it appears to be a combined passport/driving licence/phone card/payment card.
This is backed up by the novelisation which calls it a passport.
Korben grabbed the passport and read it. “Uh, yes,” he said.
“Newlyweds. Love at first sight. You meet, something goes ‘tilt,’ you
get married, you hardly know each other. Right, darling?”
As does the script
CORNELIUS: Did you get them?
David hands Cornelius two passports.
CORNELIUS: Excellent... Leeloo Dallas.
He hands it to her. The name makes her smile.
CORNELIUS: And Korben David Dallas.
Luc Besson just says (in an AMA interview) that it's bigger and better than a credit card
Q. First off, let me say, The Fifth Element is by far my favorite movie of all time. My question is, what all services come with the
A. Oh god... i dont remember! But it's bigger and better than a Platinum credit card!