Probability is either (1) experimental probability or (2) theoretical probability. When we roll a "fair," six-sided die a number of times, we may observe a frequency distribution of ones, twos, ... that is not uniform. In fact, we may get ones all the time (very rare, of course, but rare events do happen -- for example, how animals only on planet Earth have eyes to see, nerves to feel, etc.) That's experimental probability. Theoretical probability uses math to compute an "expected" probability distribution based on "fair." So, a uniform probability is used for dice, a "normal" (bell-shaped curve) is used for many events, and there are lots of others.
We know that births are not uniform across years (note: I'm a "baby boomer") or across years (early years after weddings, for example) or days (after vacations, holidays, or long snow storms, for example). However, if you assume (or presume) that each day of the year has an equal opportunity to be a person's birthday, then it has a probability of 1 out of 365 (that is 1/365) for being so. For a second (different formulas for independent and dependent events) to also happen on that same day is also a fraction (they are called "odds").
Drawing a number 6 marble from a bag of 6 marbles numbered 1-6 has a theoretical probability of 1/6. If it is returned to the bag, the probability of drawing a 6 marble again is still 1/6. And, the probability of drawing two successive 6'es is (1/6)*(1/6)=1/36. That's because probability is defined as (number of successes) / (number of all possible outcomes).
If a number 6 marble is drawn from a bag of 6 marbles numbered 1-6 and not returned to the bag, then the probability of getting, say, a 6 and a 3, is (1/6)*(1/5)=1/30.
Parental relationship may, or may not, be a determining factor. We have lots of statisticians to debate this. And, they have lots of tools (and complicated math) to calculate correlation (things happening together), but that does not establish cause-and-effect. For example, they determined that "students who eat breakfast do better in school," so we started free breakfast programs (students can even get free breakfasts during summer vacations). The outcome: students who eat breakfast aren't doing as much better in school as the used to do (because more bad students now eat breakfasts). One of the most famous (and most controversial) uses of such statistics is the FDA and others testing new treatments.
Birth dates, with parent information, are a matter of public record (and now on the Internet rather than in just paper form), so a determined, budding statistician (or student writing a Thesis) could test your hypothesis and report the findings.
Oh, I should also say that many, many "statisticians" have learned how to "lie with statistics" in order to "prove" their presumed position.
