I'm assuming you know what transference/counter-transference is... but if not. This is your introduction to it as an embodied and witnessed phenomenon embedded in your question. Also... vagueness and migration of self-narrative do not serve formulation of Borderline character structure in an of themselves, but if you mean difficulty reflecting on and making sense of internal states and volatility of self-states then yes.
The thing you might think of is. This thing that I'm feeling with this patient. What does it mean? Does this person elicit this type of reaction from other relationships outside the therapeutic one? And in that way... you take a step towards understanding your relationship with this patient as being important for modeling, problem solving, formulating, and hopefully changing the way this patient relates to all of the people in their life. You have to stay centered. The patient's pull for you to react in uncomfortable or hostile ways can be looked at as a terrible state of being in which they do not have reprieve from and have sadly learned that dramatic interplay is relating in some confused way. How would you like to be them? Perpetually confused by your interactions with people. Oscillating between anger and fear of abandonment.
So you have to model an impassive, higher form of relating. Calm, reassuring. Supporting, affirming, but not caught up in dramatic interplay with them. Let them hate you, overly identify with you, overly attach to you, hate you again, all manner of erratic confused, emotionally charged self-states. And you buddha like. Constant. Even tempered. Helping them cycle through emotional self-states in less and less intense ways.
If you hate them. Perhaps you're too attached to the notion of being loved or looked up to by your patients in the first place. Or.. perhaps you've had a bad experience with someone in the past that they recall you to and invite you to inhabit again. Whatever the case may be.
The importance of your understanding of your own internal states cannot be underestimated. But don't despair. There is a kung-fu of dealing with this sort of patient and other sorts that you learn by doing. Which you are...
It does help to be given proper training in those arts though. My program rotates us through several months of a personality disorder clinic which I think was invaluable. I'm going back this year to advance to brown belt status, after 6 months of co-leading a DBT group in my outpatient year. So.. it takes time. And patience... lots of patience. haha.