Star Trek: Discovery's season 2 retcon of Spock not only explains why the Vulcan never mentioned his adopted sister, Michael Burnham (Sonequa Martin-Green), but also his established history of not discussing anyone in his family. Originally portrayed by Leonard Nimoy, who played the quintessential Star Trek character for almost 50 years, Ethan Peck took on the role of the younger Spock for Star Trek: Discovery and the upcoming Star Trek: Strange New Worlds.
The events of Star Trek: Discovery season 2 predate Spock's voyages aboard Captain James T. Kirk's (William Shatner) Starship Enterprise by about a decade. The Vulcan teamed up with Michael Burnham, his estranged adopted sister, to save the galaxy from Control, Section 31's sentient A.I. program that planned to wipe out all organic life in the galaxy. But in order to achieve victory, Michael had to become the Red Angel and lead the U.S.S. Discovery into the 32nd century. Further, knowledge of Michael and the Discovery would again threaten the timeline so after they jumped to the far future, Spock personally redacted his sister and her starship's existence from Starfleet records. And protecting Michael included a vow of silence about her, which the Vulcan maintained for the rest of his life, even from Kirk, who never knew about Michael Burnham.
Star Trek: Discovery's retcon nicely feeds into one of Spock's particular traits in Star Trek: The Original Series: the Vulcan was notoriously tight-lipped about his family and his past. Taking the lessons he learned from keeping Michael a secret in context, Spock became closely guarded in regards to his entire family, justifying his secrets because he is Vulcan. Although he was mature and sure of himself as Kirk's First Officer/Science Officer, Spock was also intensely private and proudly logical — so much so that he often grew frustrated with his human shipmates' emotionalism, especially Dr. Leonard McCoy (DeForrest Kelley). Spock was also aware that Kirk, McCoy, and his crewmates loved to take any occasion or break in his Vulcan stoicism to tease him, which only encouraged his reticence.
Spock was so determined not to discuss his private life during TOS that in the season 2 episode, "Journey to Babel," not even his best friends, Kirk and McCoy, knew that Ambassador Sarek (Mark Lenard) and his human wife, Amanda Grayson (Jane Wyatt), were Spock's parents. Spock and Sarek were also estranged for 18 years during this time and it's possible Spock's role in Michael's fate played a part in their hostility because Sarek (James Frain) and Michael were extremely close. Sarek and Spock did patch up some of their differences but there was always a distance between them, and Michael can now be inferred as a sore subject neither would broach.
20 years later, during Star Trek V: The Final Frontier, Kirk and the Enterprise crew received another shock from Spock when a Vulcan revolutionary named Sybok (Laurence Luckinbill) hijacked the starship to fulfill his quest to find God beyond the Great Barrier at the center of the galaxy. Spock didn't immediately tell Kirk that Sybok was his half-brother, who was older than he was, and the son of Sarek and a Vulcan princess. For a time, Kirk was furious at Spock for his betrayal but they eventually worked together to find the mythical world of Sha Ka Ree and confront "God," who was an alien imposter. Sybok died saving Spock, Kirk, and McCoy from "God" and, true to form, Spock never spoke of Sybok again.
However, in J.J. Abrams' Star Trek movies, the older Ambassador Spock became less guarded about himself and his past. After he ended up in the alternate Kelvin timeline, Spock didn't hesitate to tell the younger Spock (Zachary Quinto) all of the info he needed to defeat Khan Noonien Singh (Benedict Cumberbatch). Of course, Michael Burnham was introduced as Spock's adopted sister in Star Trek: Discovery season 1 in 2017 so she didn't exist when Leonard Nimoy embodied Spock before his death in 2015. Regardless, Star Trek: Discovery's season 2 retcon in which Spock saved the galaxy with Michael and then helped erase her existence now helps explain the secrets the Vulcan kept about his family throughout the rest of his life.