Whoever owns the rights to Exalted owns the rights to Exalted; there's no "X owns 2e, but Y owns 3e." Currently it's owned by Paradox Entertainment (Paradox Interactive? Whatever), who purchased them from CCP, and who license them to Onyx Path Productions. But the owner of of Onyx Path, Richard Thomas, was the creative director of White Wolf Game Studio back when it was White Wolf Game Studio and not a just brand, and the Onyx Path freelancer pool mostly consists of former White Wolf Game Studio freelancers and employees, with the normal churn you see over time. There hasn't been any handover where the people who worked on Ex2 have been entirely replaced by a new group of people who work on Ex3, just complex legal juggling happening over all their heads.
(Though the actual developers have changed -- from Rob Hatch during 1e's first development phase before the game was released; to Geoffrey C. Grabowski for the 1e corebook and all of 1e and the 2e corebook; to John Chambers, Ex1's chief editor, for 2e after the corebook and up until near the end; to Holden Shearer and John Morke for the very tail end of 2e and the 3e core; and now to Eric Minton and Robert Vance, who were late 2e freelancers and 3e corebook writers. As I said, normal churn. Richard Thomas has been Boss of Exalted's Boss the whole time.)