A translational error. "Warring states" is "zhan guo" in Chinese, which should mean "warring kingdoms" instead.

Also, in the "warring states" period, everyone still technically were under the Zhou monarchy - until Qin wiped it out 20 years before they wiped everyone else out, but for the majority of the era every state had a vassal status to the Zhou monarchy, who only had control over Luoyang.

During the Three Kingdoms era, though, all three kingdoms did proclaim themselves to be the new, legitimate monarchies of China. First was Cao Pi with Wei, claiming that since house Liu of Han "lost their deer" - an ancient proverb: "deer" indicates the Chinese world, "hunters" indicates aspiring warlords - and it was the Cao family who helped recapturing it, house Liu had no merit to reclaim it. There was a huge ceremony in which Cao Pi forced the last Han Emperor to give him the Porcelain Stamp, the imperial symbol, and in the process the monarchy is reborn with Wei. After this, Liu Bei and his branch of house Liu also proclaimed to be a legitimate monarchy, since it was the last surviving branch of house Liu, and that Han, not Wei, still remained to be legitimate and merits the Chinese lands. After that, Wu also proclaimed their own legitimacy.

Whereas in the case of the Warring States, no one bothered to take it to the final step of proclaiming "I am the biggest and brightest, and I destine to conquer all y'all". Why? Because all this can serve to do is pissing off everybody else, and give them a casus belli due to the perceived "disloyalty" to the still existent Zhou dynasty. Qin did only after all enemies were wiped out. Whereas in the case of the three kingdoms, there was not that kind of "peer pressure": there were only 3 major powers, and Wei, to begin with, was far too powerful to care about being "ganged up" upon.