I doubt she remembered much of the 19th century, since it was all of the first month of her life. Unless you're one of those people who insist the 20th century didn't begin until 1901.
Kidding aside, it's true there was no Year Zero. But there was no Year One either. That designation was applied centuries later. In terms of human perception, the most significant event is when the first two digits change, so it makes sense to use that to mark the turn of a century.
The class of 2001 was definitely the first class of the new millennium. Fuck those class of 2000 losers!