In imperial Chinese practice, a change in ruling clan meant the end of one state and the founding of a new one. So when the last Liu emperor, Liu Xie (Emperor Xian), abdicated to the first Cao emperor, Cao Pi, in 220 AD, this meant the end of the old Han state and the founding of a new state called Wei. But the warlord Liu Bei, who was descended from a Han prince, declared the Wei state illegitimate and proclaimed himself the legitimate Emperor of Han. Several years later the last remaining warlord, Sun Quan, declared an empire of his own (because to not do so would be tantamount to acknowledging one of the others as his overlord).

"Three Kingdoms" is standard usage but it's something of a mistranslation- the word guo 国 used in the Chinese term (sanguo 三国) only means "kingdom" in certain specific contexts, and more accurately means "state" or "country." The Three Kingdoms period was an era of three emperors, not three kings, and to be an emperor meant to lay claim to all of China. So while Emperor Xian of Han was still on the throne, but exercising no actual power, the Cao, Liu and Sun warlords were all content to identify themselves as subjects of Han. Cao Pi's assumption of the imperial title meant the Caos were claiming all of tianxia for themselves, and the Lius and Suns had to either submit or find emperors of their own.