The only reason the other teens in “The Beach” didn’t realize who the Fire Teens were was because the other teens were extremely checked out on politics and the war. Like, Azula shows off her blue fire in front of Chan, Zuko is walking around with his scar and has Mai call him by his name when the other party guests were extremely focused on him and Mai, and Ty Lee not only shows off her gymnastics but also apparently tells her admirers her name. And all four Fire Teens should be extremely famous by that point(and they all have prominent roles in “The Boy in the Iceberg”). Zuko is literally the crown prince.

There isn’t really an explanation other than the teens at the beach (and at Ember Island more generally) paying absolutely no attention to politics. And maybe even if someone did suspect something, they assumed battle-hardened royal and noble child soldiers would never walk among among them like commoners.