I have a few births like this in my research and I quote one example.
My great grandfather's sister was married in 1890 and then had a daughter registered to her and her husband later that year (father / mother both on birth certificate).
Two years later, that mother then has a child registered to her by her stepmother (my great great grandfather's 2nd wife but not the bio mother of the mother in question), and although she was still married, there is no father listed on the birth certificate. All her subsequent children 2 years later and after had a father and mother listed, strange.
My ancestors made false declarations at births of illegitimate children and from my own research I have discovered that the child registered without a father was not born to the married mother as listed on the birth certificate but to her younger 17 year old sister (she was not named), that younger sister had another illigitimate child a few years later that had to be registered to her as there were no more female "volunteers" to have the birth registered to them as pregnancies overlapped. I even doubt if my great grandfather's sister knew that the birth had been attributed to her.
Now jump forward to the 1940's and my late mother kept from me the fact she had a child before marriage to my father, but you won't find that birth registered to my mother, that child was given away to married relations and then registered as their own child away from the district where the birth occurred. Then my mum had another child after me (although still married to dad but he was not the father as they had temporarily split) and that child was also given away to other married relations and the birth registered in the same way, before my dad came back on the scene
In my view a birth certificate is not worth the paper it is written on, and I have many cases in my research all backed up with certificates and information from people who knew what had happened, but there also are a few people still alive who will not say a word.
And to cap it all, illegitimate children of one family were married off to illegitimate children of other non related families.