The NYT has a thing about "What It Means To Be A Dad". It is essentially a pitch for the Dad Magazine tag on The Toast, disguised as a trend piece. That it does not mention Kindling Quarterly helps to mark the gestation period of dad self-awareness: a little under two years.
Generally, you really just do not start even conceiving of dadhood until you're in the business of conceiving. You just tune it all out. There were It Strollers and swaddling stories a year before you got someone pregnant, but you were happily oblivious to them. There were dads figuring shit out before you, too, but you don't relate. Your reference point is your own, your experience, and that includes your own childhood. Which includes, most likely and however it played out, your own experience and perception of your father.
Feeling like, being completely sure that, scarily, thrillingly aware that you are discovering and inventing parenthood as you go is an intrinsic part of the process. That's just how it is. If at some point you realize that we've all been reinventing this wheel of life since the beginning, great. But it's obviously not required.