No, the DNA isn't identical, even in twins. There is a chance that the cousins would look a great deal alike, but rarely would they look identical to one another.
It can happen that twins share one placenta, but it's rare, and it can cause problems for them. The condition is called twin-to-twin transfusion syndrome, one is small and anemic and the other is large and polycythemic.
Even with identical twins having twins, they are more than capable of having just one child at a time, and also for one couple to have fraternal (a girl and a boy) twins.
Or for identical twins of one gender in one family, and twins of the other gender in the other family. There's any combination that can come from twin partents, hope this helps.
Identical twins do share the same placenta. That's what makes them identical. One egg. One placenta.
Although in law the children of such parents will be first cousins, genetically they will be brothers and sisters and therefore share on average half their genes in common as compared with one eighth as 1 st cousins. Therefore they share more DNA than normal 1st cousins.
Yes, they would be biological siblings. But, if they're legally considered cousins, in areas where it's legal, would they be able to get married? Creepy!