Thirty or forty years ago, people apparently had much stronger identities, according to an old french psychologist I heard on the radio. People were much more inclined to think of themselves as 'a communist' or 'an artist'. It's true that these thoughts are prisons, but not having them sends people to the psychologist for other reasons. Unfortunately I don't remember what he said about these new problems.
I guess one problem with discarding your identity as a father is that you are also questioning responsibilities that are vital to your children. Another problem of going without identity is that having a sense of belonging to some community becomes more difficult. That's something I have been missing personally.