There are some theories that talk about what you are asking. Still, it’s controversial to affirm that pwBPD have no identity. I think they lack a strong sense of self, which is not the same. Still, according to the DSM IV, identity disturbance was one of the nine criteria for borderline personality disorder. That’s why many studies tried to find out the reason. The DSM 5 changed it for “Markedly and persistently unstable self-image or sense of self” For the sake of responding to your question, and to help to understand the former idea, I’ll list here some of the theories that agreed with what you affirm in your question; that pwBPD struggle developing a clear identity.
Theory of Mind One is that they lack Theory of Mind (ToM). Theory of mind is the ability to attribute mental states — beliefs, intents, desires, emotions, knowledge, etc. — to oneself and to others. ToM is necessary to understand that others have beliefs, desires, intentions, and perspectives that are different from yours. That is also called “cognitive empathy.” Impairments in the ability to infer the thoughts, emotions and intentions of others—or social cognition or mentalization—are present in a number of mental disorders. Social cognition includes emotion processing, social perception, ToM/mental state attribution, and attributional style/bias (Vaskinn et al, 2015). It seems that interpersonal dysfunction can be an effect of irregularities in social cognition. This perspective assumes that because pwBPD suffer from hypervigilance, they are extra sensitive to social cues overly attributing intentions, and therefore, committing overmentalizing errors.
In summary, what it means is that pwBPD are so concerned about how to be safe (and attached) that they put all the energy in trying to understand others, and since they are most often wrong, they can’t develop an understanding of who people are including themselves.
Dissociation pwBPD have abnormal autobiographical memory functioning; for instance, they show memory biases in autobiographical recall. Their memory deficits have been connected to problems with their narrative selves. Memory problems have been examined in the context of dissociative experiences, where the subjects may have large gaps in their memories, may be bothered by how much they have forgotten and may not remember what they did or said when angry (Gold et al, 2017).
Dissociation involves the disruption of consciousness, memory, identity, or perception of the environment. pwBPD normally suffer from pathological dissociation, with dissociative amnesia leading to characteristic mental holes or blind spots in their autobiographical memories, which interferes with a narrative self.
Additionally, pwBPD processing of self‐relevant information is influenced by impulsivity, emotion‐focused coping strategies, and tendencies to rely on an external locus of control, which causes the characteristic mental holes or blind spots that can show in the autobiographical memories.
This theory assumes that the pwBPD can’t develop a coherent narrative of the self, and therefore can’t form an identity.
Fragmentation The easier to understand is the one that talks about how pwBPD have a fragmented self due to the splitting between loving and fearing the caregiver.
This is another type of dissociation that confuses the child on who s[h]e is, since s[h]e experiences such polarization of emotions and conflicts with his/her need to attach. Therefore, the emotional states grow separated stoping the person to develop a sense of unity, and therefore, of an identity. ——- Of course, everything is connected with failures in executive functions since the attachment rupture that may be common to almost all pwBPD highjacks the development of the prefrontal cortex.