Some families do have several sets of identical twins in the family. It can appear they actually "run" in those families. What causes the fertlized egg to split? It does happen by chance, but can it be genetic for some? People are still learning about identical twinning and some believe that identical twins aren't truly identical.
Fraternal twinning is only passed through the mother. I'm a fraternal twin but my dad is also a fraternal twin (male and female in both cases). I'm not sure if fraternal twinning even runs in my mom's side. It would appear my dad had something to do with it. But, men have nothing to do with women releasing more than one egg, though.
I can see why people would be confused about all this. There is still a lot to learn about twins.