While the math is clear, I'm a bit annoyed by the label "paradox" as the whole setup is too simplistic and reductionistic.
The actual chance of being in the same room with someone who shares your birthday needs to include other factors like your socioeconomic background, the cultural environment you are in, your present location, and certain historical facts.
Without having done the math, I'm fairly certain that a member of the baby boomer generation in New York has a higher chance of meeting their birthday sibling than a 12-year-old in a rural part of Australia.