Filial Piety
Classism and family ties are recurring themes in Soul Society dynamics. While many of the featured shinigami hail from slum-like districts and lack blood relatives, Byakuya stands at the height of privilege in the Kuchiki clan. Expectations of proper conduct and strict adherence to the rules fell on his shoulders, shaping his growth and also highlighting his drive to find his own freedom.
Youthful Passion
In the flashback arc Turn back the Pendulum, Kubo graces us with the glimpse of a shinigami student who looks like a prince, but has the temper of a steam whistle. It's hard to imagine that Byakuya's face muscles were ever that flexible, but at least we know that his uptightness was developed rather than innate.
This revelation makes a powerful statement about nobility in the series. Characters like Renji and Rukia who always struggled to make ends meet largely kept the fiery parts of their personalities, whereas Byakuya had to suppress it as he matured. No one comments on the change in his demeanor or if anything specifically triggers it, but I get the feeling his no-nonsense attitude came about in atonement for past "mistakes".
Tradition vs. Freedom
I swore on my parents' grave that this would be the last time I ever broke the rules.
Two events, which will be expanded on in the Relational section, evidence that a past Byakuya could defy one of the cornerstones of his identity: his marriage to a woman with no status whatsoever, and the adoption of that woman's equally status-less sister. We only hear about the acts from Byakuya's unembellished descriptions, but if he is any example of how legalistic the clan is, the backlash must have been quite significant internally.
The Kuchiki name taught him to be loyal and honorable, so for him to act against clan wishes indicates the resilience of his character to choose love. However, just because he could resist clan laws, he would never try to escape them or sever himself from them entirely; that legacy is too much a part of him.
One of Byakuya's most interesting character points is his growth in understanding how to cope with his noble responsibilities and his personal ones. It's a hilly landscape: rising against the mold, conforming to it, then just breaking it entirely when he sets his priorities straight about what "family" really means. The name isn't important; it's the people you share it with that matter most.