The way in which a child modifies her own sense of self to incorporate some ability, attribute, or power she sees in others, usually a parent, is known as:

Psychoanalytic Identification

Explanation:

Identification is an mental process whereby a person unconsciously tries to modify their personality to behave the personality of another person who they see as an example.

Psychoanalytic Identification is the unconscious mental process whereby a person tries to change their personality to behave like the personality of their parents.