This pretty much depends on your definition of "in some form or another." Egypt could very easily compete with China for that spot on the list. The biggest problem I'd have with it is the definition of "country" as it implies a kind of statehood that the Chinese of 5,000 years ago couldn't even dream of, nevermind the Chinese of 2,000 years ago or the Chinese of 200 years ago.

This claim is based more around nationalism and an attempt to square modern concepts with historical ideas of a Sino-Centric worldview.

In the narrowest sense, "China" of today began in 1949, or 1912 if we're being generous, starting with the Communist Revolution or the Republican Revolution, respectively. In a slightly looser definition, we could say that these modern incarnations of China inherited the dynastic tradition that started c.200 BCE when the Qin Dynasty emerged triumphant over the Warring States of Ancient China. This is a marked line in the sand in Chinese history because dynastic China was a fundamentally different place than ancient China, possessing some of the characteristics and hallmarks that we would come to associate with China's historical dynasties: namely Confucianism, Taoism, Legalism, and the Mandate of Heaven. These four concepts did exist prior to the Qin Dynasty, but they were largely nebulous. The Qin standardized a lot of these strains of thought that the Han Dynasty, which lasted for four centuries, reaped the benefits of and solidified into a Classical Period of Chinese history. Future generations of China would look back to the Han Period as a sort of Golden Age.

Anyway, prior to the Qin, there was the Warring States, and the victorious Qin weren't even regarded as "full Chinese," whatever that meant at the time, probably because they were the western most "state" and intermingled with non-Chinese peoples more regularly. The Zhou Dynasty, allegedly ruling over the other states, was in a long period of decline, and prior to them was the Shang and the Xia Dynasties, both semi historical and lost mostly into the mists of time. Combine that with the fact that to mark his new reign, Qin Shi Huangdi, the first Emperor of Qin Dynasty, held a massive book burning to eliminate competing histories and philosophies, and you have a pretty sharp historical cutoff between "historical China" and "claimed China."

Sure the "5,000 year" claim has some merit in the sense that the area that is currently within the borders of the claimed People's Republic has been inhabited for that long and even longer, but it's a confusing claim to even begin to parse apart because the nature of what a country is isn't a standard measurement we can go out and mark, and the nature of history itself tends not to draw sharp lines.

Sources:

China: A History by John Keay