The short answer, obviously, is “Because JAY-Z and Beyoncé did it.” Specifically, the musical power couple attempted to trademark the name of their first child, Blue Ivy Carter, back in 2012. (The petition actually failed in the U.S., although they did get protections in the EU.) At the time, everyone assumed this was the precursor, to, like, Blue Ivy-branded baby formula and diapers and whatnot, but the couple have always held that they were just looking to stop scammers from capitalizing on their daughter’s distinctive name. (A stance backed up by the fact that they’ve never actually attempted to sell any Ivy-labeled products—a decision that actually made it much harder for them to get the U.S. copyright on the name, because part of the trademark requirements state that the property in question has to be being used in a for-profit capacity.) Bey and Jay followed the decision up with similar efforts last year, reportedly filing for Sir and Rumi Carter to get the same legally protected treatment.
Prior to that, there doesn’t seem to be much earlier precedent for celeb-baby-name IP; we were honestly shocked to learn that Gwyneth “Goop” Paltrow had never tried to pull this shit with Apple, etc. (Victoria and David Beckham do appear to have trademarked all four of their kids’ names, but that came in a post Blue Ivy-world.) Still, they all pale before the original American “trademark your kid’s name for fun and profit” story: If you’ve got a kid named Wendy in the States and you want to name a restaurant after her, you might want to watch your back.