Psychiatry would be the specific medical field, but that probably isn't what you mean.

Abnormal / Clinical Psychology.

I guess counselling...

The usual suspects...

I came at this by way of philosophy. Personal identity is a known area over there. Lots of that was about continuity more generally, though. So... The ship of Thesus has many repairs done over the years so that not a single nail is left from the original... But the ship of Theses persists. As do we. Despite every single cell in the body being replaced after however many years.

So then I looked at the notion of a 'self' and what a 'self' was supposed to be in philosophy / psychology. The idea of 'multiple selves' and how a self was supposed to be different from an alter (e.g., why dissociative identity disorder rather than multiple personality disorder? what is the difference supposed to be? what is a self / personality / identity?).

I developed a bit of a hobby interest in 'self psychology' which is a school of psychoanalytic thought founded in the work of Kohut... I liked what he had to say about the self... About disorders of the self...

I don't know what you mean about fields of study... You could study this sort of thing from within psychology or philosophy... Counselling... Education... You probably don't want to go do medical school... But maybe you do...