Your question got me thinking, and I realized my girlfriend and I also have a strange birthday coincidence. So, first of all, the odds of sharing some weird connection with any random person are probably quite high.
From experience I'd guess around 20% or so, no way to seriously calculate that, but no matter what it exactly is, just want to be clear having a special weird connection means nothing (though it is fun).
Although the question is about birthdays, the "birthday paradox" isn't really relevant here.
It's about how many random samples you need to take before you expect at least two samples among them to be equal (a collision).
[Edit: Jeromy gives a figure based on actual data, which turns out to be about 17%.] According to wikipedia, 33.2% of married couples in the United States differ in age by less than one year.
Then, something the other didn't take into account, looking at birth rates per month we get a nice overview (it's caused by things like people being off thus having a lot of free time on hand 9 months before the months in question), next dividing that percentage by the number of days in that month.
Next one should figure out what the chance is of being born in the same year.
The larger you cast the search, the more possible bases there are for finding coincidences.
In general, the more you look for them, the more you will see them.