It's sometimes called the Birthday Problem to avoid the paradox label. (See, for example, the wikipedia article.)
The paradoxy bits of the problem are the things that people find confusing. "Only 23?!" or "So if there are 23 people in a room there's a 50% chance that one of them will share a birthday with me?"
The paradoxy bits of the problem are the things that people find confusing. "Only 23?!" or "So if there are 23 people in a room there's a 50% chance that one of them will share a birthday with me?"