Every single day we talk about ourselves, saying "I did this," "I did that," "I am," and "I'm not," but we don't usually think about what "I" means. In Pygmalion, Shaw forces us to think this through. Some characters want to change who they are, others don't want to change at all. Things get even more complicated when identities are made up, constructed. The play wants us to ask ourselves what I really means to think about different versions of the self, and whether that self can ever really be changed.
Eliza's identity is, from the very beginning, fixed. It is only her circumstances that change.
Higgins's reluctance to change reflects a deep insecurity in regard to identity, an insecurity fostered by his own life-changing abilities.