Final answer:
The theory in question is Freud's Psychoanalytic Model, which includes the identification with the same-sexed parent during the psychosexual stages, notably the phallic stage, of early childhood.
Step-by-step explanation:
The theory that recognizes that a child needs to identify with the same-sexed parent, and where an older sibling could serve in this role, is D. Freud's Psychoanalytic Model. Sigmund Freud's theory proposes that personality develops during early childhood through a series of psychosexual stages.
This includes the phallic stage, where children identify with their same-sex parent. Alternatively, the role can be assumed by a significant figure, such as an older sibling. Freud's theory suggests that early childhood experiences and the handling of psychosexual stages have significant implications on adult personality and behavior. On the other hand, Erikson's Psychosocial Theory of Development diverges from Freud by emphasizing social rather than sexual nature in development, suggesting that personality development occurs throughout the lifespan and is influenced by social interactions and tasks.