I do think there is something to this, though I think part of it is probably cultural rather than strictly physiological. As in if your mother had a certain birth experience or birth tends to happen a certain way in your family, that's much more normal and expected for you, so history tends to repeat itself. Our beliefs are very powerful and what's normal has huge variation, but stories passed down about birth can have a real impact in your life if you buy into them.
In my case, I don't know anything about any other women in my family, but my daughter's birth (my first) was very similar to my mother's experience with me (her first and only). We both went on the earlier side. I was born around 38 weeks and my daughter was 37+5. Both completely natural, started with waters breaking in the middle of the night and labour was really manageable. I had a home birth, but I was really comfortable until the end when I was pushing. With me, my mum's waters broke about 1am and she was at a cafe on the way to the hospital waiting while my dad had breakfast at 9am. They got to the hospital a bit after that and I was born at 1pm. So both really manageable and straightforward and almost exactly the same length with waters going first.
It was also our first babies, we both wanted a natural birth, we were close to the same age, equally healthy and physically active all through pregnancy, so very similar in a lot of ways. But realistically, I think a lot of that probably also had to do with just being raised to know my mum had a fairly easy, straightforward natural birth and expecting the same. She has always told me the story of stopping to have breakfast on the way to the hospital and the waitress actually dropped and broke a coffee mug when she told her she wouldn't be eating because she was in labour! So I think it was just the norm and that's what I expected. Similarly, my husband's family has had almost all natural, easy births, many of them home births (including our daughter's too), and his granny had 4 of her 5 babies at home. It's just the norm. So I think there is something to that, in addition to there being physiological similarities. Also, realistically, most first babies are born at 41 weeks if left naturally to come when they're ready, so it's not so much they are 'overdue' as on time based on the averages, so to an extent it just skews that way anyway. The ones that come early, like me and my daughter, are more the exception, even if we did seem to follow a pattern.
. My mum, grandma, aunts all needed to be induced. My mum gave birth 4 times, I was the only girl and the only one who didn't require forceful eviction 
. My first was induced (boy, born at 42 weeks), 2nd came spontaneously at 41 weeks. 3rd was induced but not for being post dates so I wonder whether she'd have come on her own if she'd had longer (she was born at 40+5). 
