I heard of twins both wanting to join the army, but because they were born before & after midnight and had different birthdays they had to be in different intakes.
The doctor would - of course! - enter the births on the day they occured. The fact they are twins is irrelevant, if one is born on the Monday and one on the Tuesday that's how it is, whether the time difference is a couple of minutes or a couple of hours.
I have also heard of a midwife moving the bed between twin births for better light so the babies actually ended up being born in different counties!
The twind I know would have been delighted to have had a different birthday from their twin ,I'm sure, they get fed up with sharing everything.