Bob is older than he looks.
Specifically, even though Bob looks like he's in his twenties or thirties (it's hard to tell), he was actually born in Europe about ten thousand years ago, long before the first word was ever written down.
By coincidence (or not), on the night (or the day) of Bob's birth, something unusual happened in the sky. Several people noticed, and thought that it was an important omen. So, Bob grew up knowing that he was born on the day that this particular event happened.
Furthermore, with the development of astronomy and computing, Bob has finally been able to determine the exact date that this event occurred, and thus, his date of birth.
What could the event have been?
Answers to questions I've received in the chat room:
Does it have to be a one-time thing or is it something that could have happened repeatedly? It can be something that happens repeatedly. A total solar eclipse that happened around that time would fit all my criteria.
Just how visible was the event? Visible enough that someone living in Europe ten thousand years ago plausibly could have noticed it and pointed it out to other people. Not necessarily visible enough that multiple people would have noticed it independently.
How precisely does Bob need to be able to determine the time that the event occurred? Down to the day. Some astronomical events (like supernovas) can't be calculated back with that amount of precision, so those wouldn't be suitable. Other events (like solar eclipses) can.
How much information does Bob already have about the date? I figure that Bob knows the season in which he has born, and he has also calculated the exact number of years since he was born. So he already knows the season and the year; he just wants to pin down the exact day within that season.
Are you looking for an actual historical event? Yes, I'm looking for the date that an individual, real astronomical event actually happened.