Given that we're dealing with anime, and not binary numbers, as such we must treat the Pokemon as if they are living creatures (to an extent), I'd say equivalent wise Kukui's Incineroar did reach 0 HP with perhaps Torracat having 1 HP left, then when it evolved, Kukui's Incineroar recovered just enough to gain 1 HP, while the stress of evolving and expended energy caused Ash's Incineroar to go down to 0 HP, and because it suffered more injuries, it would take longer for Ash's Incineroar to regain that equivalent 1 HP back while Kukui's Incineroar just simply was knocked out but was more experienced to recover just enough to get back up soon after.
We have to assume that just because a Pokemon recovers doesn't mean that it can still be used in battle if the referee calls it, it needs to not be used any further at least until it properly is taken care of by Nurse Joy or perhaps after a long period of time.
It's hard to know for sure if Pokemon can recover their full strength on their own, but I assume it would make sense. I just think the Pokemon Center merely speeds up the process, or handles more serious injuries with of course whatever magical medicine they use that can technically fully heal a Pokemon that was on the brink of death. The exception that we've seen is Ash's Pikachu nearly dying in DP, in which magical medicine wasn't enough to fully heal Pikachu, but it did seem to confirm that Pokemon naturally can recover from serious injury if given enough time, though probably needing actual medicine to FULLY recover. The exception of course was Stoutland but I would assume age of the Pokemon plays a big part on the presumed "immortal like" ability Pokemon have or perhaps serious conditions at play (such as being poisoned too long without treatment, which would somewhat contradict the games where Pokemon can recover while being in Poke balls).