YES LV has serial numbers and date codes, but never a ''model number''. Only the bags from the French luggage company lack a number. All other bags will have them.
That said, your serial number doesn't really line up with the LV format. There should be two letters, followed by four digits. The letters are the factory where the bag was made, and (if you bought the bag after 2007), the four numbers are the week and year it was made. If you bought the bag between 90-07 the first and third digits are the year, and the second and fourth are the date. The first two digits should be higher than 80.
The only way to know for sure would be to take it in to a LV boutique and have it verified.
Assuming it WAS actually purchased at the LV boutique and said gift-giver is telling the truth, it should be real.
If it was bought on 'sale', wholesale, from an outlet store, 'last year model' sale, etc. then it is FAKE! LV does not have sales. Ever. Never ever.
If you want to post a few pics of the bag and the tags I can take a look.