Which theories of personality development (Psychoanalytic, Humanistic, Social Cognitive, or Trait) would you use to explain the differences between how Greg and Mike turn out in Life on the Color Line by Gregory H. Williams?

Of the four personality theories, it appears as if the trait theory best explains the differences between Mike and Greg. Major theorists of this perspective are Hans Eysenck, Raymond Cattell, Robert McCrae, and Paul Costa. According to Hans Eysenck, the three major dimensions of personality are extraversion-introversion, emotional stability-neuroticism, and psychoticism.
So, two siblings from the same dysfunctional family can exhibit varying degrees of stability and instability. In the story, Greg (the narrator) and his brother, Mike found that they were on their own the minute their mother deserted the family. Their father, Tony, was not much of a solace; heartbroken and hurt by his wife's betrayal, he took to drinking heavily and to taking up with the wrong crowd. Miss Dora, an older lady, adopted the boys, but Tony rarely contributed to their care. Instead, he forced Mike and Greg to take up odd jobs to support all of them.
Greg relates how, for his first job, he had to pay a commission to Tony out of the three dollars he made "hustling" coal for a drug dealer. Every job Greg and Mike had was usually arranged by Tony; because of this, Tony expected both Mike and Greg to pay him commissions from their weekly pay. Tony usually drank away any money his young sons gave him.
Because of Tony's irresponsibility, the brothers lived in abject poverty for most of their growing up years. To make matters worse, Tony treated his two sons differently. While he encouraged Greg to pursue his dreams of becoming a lawyer, he treated Mike as a younger version of himself. For his part, Mike looked up to Tony and sought to emulate him.

He drank, gambled, had an active sex life, and mimicked the lingo and style of the small time hustlers, pimps, and thieves who befriended him...He reveled in tagging along with Dad on his rounds to gambling joints, bootleggers, and Muncie's whorehouses. Mike willingly absorbed all of Dad's lessons.

Mike was emotionally tied to Tony, while Greg sought to distance himself from his self-absorbed father. In Hans Eysenck's theory, Mike would be considered a highly neuroticistic extrovert; to survive in a dysfunctional relationship, he was willing to repress and to ignore his pain. On the other hand, Greg was not so willing to give Tony a pass for his abusive behavior. Greg would be considered less neuroticistic in outlook than Mike.
We can see the difference between Mike and Greg in the Christmas card episode. One Christmas season, Tony procured some Christmas cards for Mike and Greg to sell. The idea was that the boys could earn enough money to buy their own Christmas presents. However, Greg knew better. He refused to hand over any money to Tony; with the proceeds from the sale of the cards, he purchased a pair of shoes for himself. When Tony harangued Greg for not handing over his "investment" and his commission, Greg nonchalantly asked his father whether he would prefer to have his shoes instead.
This made Tony angrier, and he eventually threatened to leave the boys. For his part, Mike begged Tony to stay, promising that he would do whatever it took to make Tony happy. That Christmas, it was Mike who did without a Christmas present. So, the trait theory best exemplifies the difference between Mike and Greg; the theory hypothesizes that everyone displays varying degrees of stability and instability within specific environments (whether dysfunctional or not), and this bears out in Mike and Greg's story. While Greg eventually realized his dream of becoming a lawyer and then a law professor, Mike was left to pick up the pieces of his life in Indiana after being blinded in a shooting in an Indianapolis bar.
http://webspace.ship.edu/cgboer/eysenck.html

https://www.verywellmind.com/personality-perspectives-2795950