One consideration with Egyptian dynasties is that there are just so many of them (37 by one count, spaning about 3000 years), it just makes more sense to group the dynasties together in larger period (eg. Old Kingdom, Middle Kingdom, New Kingdom) broken up by intermediate periods when the power of the pharaohs was much weaker.
Note: some later dynasties do have common names like the Macedonian dynasty and the Ptolemaic Dynasty
(Toby Wilkinson's 'The Rise and Fall of Ancient Egypt' for the dynasty count)
I can't say much about the Egyptian case, for example, I don't know if Egyptians during the Ptolemaic Dynasty refer to themselves as Ptolemaic Egypt.
However for the case of China, the dynastic names ARE the official name of the country. In those times, the term middle kingdom refer to rather generally the geographic region where China is situated. Politically, the country name is the dynastic name. For this reason, it would makes sense to refer to previous dynasties by those names.
A large number of founders of dynasties have held lordship over some region. When a new dynasty is established, the dynastic names themselves are usually, not always, is chosen from the name of this region. In other cases, the founders were regional warlords who operated at some regional base, and when they claim imperial titles, they establish a kingdom based on the name of this regional base.
To make an example, it would be like in England, the Duke of Kent started a rebellion and conquered the whole of England. He would then start rename the Kingdom of England into Kingdom of Kent, and Kent would then be the dynastic name, until some one else comes along, say Earl of Gloucester, who then rename Kingdom of Kent into Kingdom of Gloucester.
The *Egyptian dynasties were named by a much later historian, Menetho. Ramses II did not know he was a pharaoh of the eighteenth dynasty. The Chinese dynasties, on the other hand, were all self named. It is also worth noting that the division between Chinese dynasties is always very stark, such as civil war or invasion. This is not the case with Egypt's.