Does Honda Fit 2009 has a main drive serpentine belt?

The serpentine accessory drive belt is indeed a service item .. I feel sure if you ask to see the old belt the installer will be glad to show you why it was replaced .. Usually these belts begin to crack on the inside while the outside of the belt looks just fine .... I do this for my 'regular' customers at the shop quite frequently, however I do show them the old belt when they get ready to pick up their vehicle and they're always appreciative that I did the preventive maintenance for them .. I would rather think the belt installer did you a favor as opposed to installing a part that wasn't needed.. That said, IF you aren't a 'regular' customer of this facility and don't really know the folks enough to trust their judgment - you should have been informed and your permission given before the belt was replaced.

You are correct, you do not have a timing belt its a chain. The serpentine belt is in no way a timing belt. With that said, the serpentine belt does need to be replaced from time to time. Its usually inspected during a service and if cracked it should be replaced. Either way if you did not approve the repair then they should not have replaced it.