Yes, they theoretically share a common past, but the difference is in what they can do going forward. Kelvin branched off in a new direction so that it can build its own new history going forward. Its stories don't have to lead into the events of TOS or TNG or the rest, and can totally contradict everything that happens going forward from the opening moment of the first movie. Vulcan can be destroyed, Kirk can get the Enterprise much sooner, Amanda and Pike can be killed, etc. But Discovery is an in-continuity prequel. Its events have to be consistent with the future we know. The Sarek in this show is the same Sarek who will have a heart attack on a journey to Babel, implore Kirk to bring Spock's katra back from Genesis, and have a mind meld with Picard as he suffers from Bendii Syndrome. All of that is in his future and he can't be written in a way that contradicts that. So the '09 movie could show Sarek reconciling with his son a decade earlier than he did in TOS, but Discovery cannot do that, because it's supposed to be something that happened in the past of TOS. Whatever Sarek does or experiences in this show has to be something that's compatible with his life history in TOS, the movies, and TNG.
Well, it depends. As an industry-insider usage, "reboot" means any revival of a dormant media property. But in fan usage, ever since the 2004 Battlestar Galactica reboot, it's been assumed to mean something like that show, a completely reinvented version that's out of continuity with what came before. By that definition, for instance, Spider-Man: Homecoming is a reboot because it starts over and ignores the previous Spider-Man movies, but Fantastic Beasts and Where to Find Them is not a reboot because it's in the same continuity as Harry Potter. The new MacGyver is a reboot because it starts over completely with a new version of the character, but the new Doctor Who is not a reboot because it's meant to be a continuation of the original series. Then you have things like the Kelvin films and the past few X-Men films, which use time travel to have it both ways -- they're part of the same narrative continuity but they alter its events, so it's effectively a reboot while still being a continuation.
So anyway, by that terminology, Discovery is not a reboot, it's just another prequel series like Enterprise was.