@Caimarie,
It's a coincidence.
My nephew and I are born on the same day. So are my cousin and her nephew (different day but theirs are identical to each other).
This image shows the likelihood of anyone having a particular birthday. You would think it would be 1 out of 365 (and with February 29th being even rarer). But there are actually days which turn up more often as birthdays.
Looks like June 6 is kind of in the middle, maybe lower middle, when it comes to likelihood. June 6 puts conception right around Labor Day/end of summer/beginning of school for a lot of people.