But there is no dad role any more than there's a mom role. There's all kinds of dads and moms out there. So, I guess I'm asking, what is the advantage of calling yourself a dad over a mom?

I can see how calling yourself a dad helps challenge people's expectations of your relationship with your children and can force them to open their eyes a little to the fact that the maternal parent doesn't always need to be the primary parent. On the other hand, I can see how calling yourself a mom and refusing to do the stereotypical mom things can also accomplish this.