Not really the same thing at all, no.
The only tie between Sumer and America is one of remote influence ; Sumer influenced Phoenicia (possibly with other links thrown in) influenced Greece influenced Rome influenced England, a part of which went on to become America.
In China's case you can trace a much closer line - Qin China became Han China became Three Kingdoms China became Tang China became Song China became Yuan China became Ming China became Qing China became Republican/Warlords China became Communist China (probably forgetting a step or three along the way).
A much closer comparison would be the evolution between, say, the post-Charlemagne kingdom of the West Franks and today France. Culturally enormously different, but you can definitely trace a relatively unbroken line as a nation - Carolingian France became the medieval Capetian France, then Renaissance France under the Valois and Valois-Angoulème branches of the Capetian house. Then came the Bourbon and early modern France, then the mess of rapidly changing governments that followed the revolution (First republic, first empire, restored monarchy, second republic, second empire, third republic, fourth republic, fifth republic).
Similarly with England. The culture of England today has very little to do with that of Anglo-Saxon England - but that changes nothing to the fact that England is well over a thousand years old by now (although it nominally doesn't exist as a nation anymore, of course, and the Scots and various others would be quite cross at people who claimed United Kingdom = England)