On my tree I share my birthday with a 1st cousin 4 times removed, born in 1859, and that is all out of a very large tree. My mother, however, had the same birthday as her grandmother, and it was the date she was due to have me, only I decided to be nearly two weeks late.
I do, however, share a birthday with a friend's sister, including being born in the same year. We are not at all alike!