"China" or "Middle Kingdom" as it's called in China, was historically a fairly vague concept. It was popularized in the 20th century by Sun Yat-sen to refer to the 56 ethnic groups the republic inherited from the Qing empire (Han, Manchurian, Mongolian, Tibetan, Uighur being the major ones).
The 5000-year history refers to the lineage of Han culture in particular. For example, in Japan where most if its Sinic influence was before the republic, Chinese medicine is called Han medicine (Kampo 漢方), Chinese writing is called Han script (Kanji 漢字).
I understand what people mean when they are calling China 'young.' My comment was a bit of a snarky commentary on how China seems to pull out this "we're a 3000-year-old country" line whenever someone criticizes them. Such a response is of laughable logic since the government -- which is what most people are criticizing -- is far younger than 3000 years.
(I keep saying 3000 years, because I'm pretty sure the number I've seen used on multiple occasions is 3000, not 5000. I'm not sure from what point in time -- other than the obvious '3000 years' -- they are calling the start of this period to which they are referring, though.)
3000 years is about how long China existed as a state that passed from one dynasty to another in recorded history. 7000 years is how old the civilization is according to archeological digs, though for a long time they thought it's 5000 years, so 5000 kinda got stuck as a catchphrase.
But yeah, pulling the age card as a comeback is pretty lame sometimes.
China seems to put so much emphasis on their being a '3000 year old country' so I think we should take them up on that and stop calling them 'young.'