If you're locking the area to a fixed area out of the 365 discrete areas, the birthday paradox isn't applicable anymore. Birthday paradox only says something about whether a pair exists among all areas.
So you're in a room with N people and learn that somebody else shares your birthday. Can you conclude that is this likely that other people in the room also share the birthday with somebody else?
If you already assume the room is a representative sample from a population where birthdays are uniformly distributed, it doesn't tell you anything.
But if you aren't certain about that, it makes it less likely that everyone else with your birthday has been ritually murdered or otherwise systematically excluded from the room, and slightly more likely that you're at a convention dedicated to people with your birthday.