I think the emphasis on other answers has been in solving the abstract puzzle that was presented (which is very reasonable on this forum). In an interview situation, if you were able to come up with these different possible explanations, then selecting the "best" explanation (assuming you get one chance to answer) would probably depend on the nature of the interview.
The time zone answer is an appropriate choice for any company which engages in business across multiple time zones (demonstrates awareness of the issues that time zones can create), and for certain companies it may even earn bonus points if you can point to a specific location where this could have occurred: "Foo was born in X while his parents were travelling towards Z on a boat, Bar was born in Y 5 minutes later" where X Y and Z are locations that the company does business.
The mixup identity answer is the answer I'd go with if I were interviewing for an editorial/newspaper job because it demonstrates a focus on the precise wording of the question ("Foo celebrates his birthday on X" instead of "Foo was born on X")
And finally, if I were interviewing for any position related to a religious organization which defines life as beginning at conception, it would be very hard to go wrong with the conception answer.
Personally mixed up identity is what I believe to be the generic "most likely" answer to this puzzle (in that there probably exist more people who have been born under those circumstances than the time zone answer, and it doesn't require playing with the definition of life), and it seems logical that if the puzzle is worded as "celebrates his birthday on..." there's a reason it didn't simply state "was born on," so if a business doesn't meet any of the criteria above I'd definitely go with mixed-up identity. Even more so if it's a stock analysis or prediction of any kind job where best guesses often have to be relied upon because you cannot know something with certainty.