Happy Birthday, Lincoln and Darwin
I knew that today was Abraham Lincoln's bicentennial, and also knew that Charles Darwin's 200th birthday was coming up, but I didn't realize until today that both men were born on the same day 200 years ago. I haven't rigorously checked all possible other dates, but considering that the biggest pair of twins I can think of doesn't really come close, that's got to be the biggest one-two punch of famous people born the same day of the same year.
This guy did some research and came to a similar conclusion; the one that strikes my fancy the most is George W. Bush and Sylvester Stallone on July 6, 1946. He also came up with some interesting coincidental death dates, although this discussion points out that Shakespeare and Cervantes didn't actually die on the same day --- Cervantes died on April 23, 1616 of the Gregorian calendar (which we use today), while Shakespeare died ten days later on April 23, 1616 of the Julian calendar, which England still used at the time.
The same discussion also brings up pairs of people who died on the same day non-coincidentally, such as Jim Bowie and Davy Crockett on March 6, 1836 at the Battle of the Alamo.