File numbers are indeed assigned with an F and then a string of numbers for Family class applications.
However, IRCC does issue each file with a temporary number as soon as it's received, and then a permanent file number when it's been accepted for processing. I don't know what the temporary number looks like, so the number the agent gave you could just be a temporary number; or, and I'd think this is more likely, it is your permanent file number.
Are you able to use that number to look up the status on ECAS?
https://services3.cic.gc.ca/ecas/security.do