Netflix series conveys important contextual information, and often replicates scenes from both of the first two films. Still, a firm understanding of the backstory does indeed matter, especially when discussing why Robby Keene (Taylor Buchanon) trains with Daniel LaRusso (Ralph Macchio) — the longtime enemy of his father, Johnny Lawrence (William Zabka).
Cobra Kai season 1 immediately makes it clear that Johnny and Robby don't have a strong father-son connection. The series begins with Johnny musing about the past, and struggling to maintain healthy relationships with people in general. After a chance encounter with Daniel — now the successful owner of LaRusso Auto Group — Johnny decides to re-open the Cobra Kai dojo. Decades prior, he was forced to leave by his mentor John Kreese (Martin Kove) after losing to Daniel in the 1984 All-Valley Karate Tournament. Cobra Kai season 1 frames Johnny as a misogynistic bigot who's stuck in the past, and a father who pays little attention to his son, Robby; a teenager who needs a strong male figure as a system.
In Cobra Kai's past.
Robby trains with Daniel for practical purposes, but the character psychology implies that he's looking for recognition from Johnny. In Cobra Kai season 1, Robby ends up squaring off against the Cobra Kai student Hawk (Jacob Bertrand) during an official competition match, and becomes the target of a "no mercy" cheap shot. And so the narrative reflects what transpires in The Karate Kid's final act, which sees Cobra Kai students Bobby and Johnny intentionally injuring the Miyagi-Do student Daniel — illegal and underhanded moves used to gain an advantage in the competition. With Cobra Kai season 1, the subtle twist is that Johnny feels guilty about his teaching methods; he fully understands that Hawk's actions reflect Kreese's lasting influence on Johnny, which he has ed down to his own students. Even though Johnny wisely apologizes to his son Robby, his overall actions suggest that he's not quite ready to man-up and fully take responsibility.
Cobra Kai season 2 uses Kreese as a way to initially divide the central characters, but also as a way to bring them together. Robby essentially becomes part of Cobra Kai season 2, the evidence suggests that Johnny and Robby will resolve they differences moving forward, and may end up even fighting together as of Miyagi-Do, based on the fact that Kreese is now in control of Johnny's dojo.