If God is dead then everything is permitted, they say. But would it not be more correct to say that if god is dead then nothing is taboo? Frankfurt wants to say that if we have no restrictions placed upon us, then we cannot begin to form our identity. His idea is that we need a starting point from which to continue, this starting point in some way 'showing us the way' or some such. So if this initial lesson is not given then we are not capable of becoming a person.
But this seems to assume much, particularly that the intial limitations placed on us are of such an extent that they actually do constrict our options. And not just constrict but also reduce them in an actual way. I bring this up because once there are no limits we would have infinitely many choices. And even if our choices were constricted by half, we would still have infinitely many options. So indeed it would seem that in order for our actions to be limited/reduced in a real way, so that we actually had fewer choices, they would need to be positively, rather than negatively, described (for this is the only way to say that 'this is all the options available').
No, I don't think that any system does this, let alone the system of Dostoevsky's soon to be ex God. And thus, despite God's ability to forbid a large number of choices, the removal of these restrictions does not actually enlarge our possible options. So such a removal could, I believe, in no way lead to an identity crisis, as Frankfurt would have us believe.
So the only way one can come to a crisis of identity is by finding one's self required to do mutually exclusive things. Then one is forced to choose.