I'm an identical twin but have both identical and fraternal twins on my moms side of the family. Fraternal twins can run in the family but hyper ovulation is a gene of releasing 2 eggs in one cycle that can be passed down as well :)
Your dad's input does not make any difference, not only is identical just luck of the draw, but nothing about twinning comes down the dad's side. But you being a fraternal twin (which comes from your mom's side) does. You have a greater likelihood of having twins (fraternal) than a woman who does not have fraternal twins in her mother's side of the family. It comes from having the ability to pop out two eggs in one cycle. Your mom might have passed that down to you.