I understand that all of those are vague terms at the best but typically in everything from modern pop culture to politicians in India and China portray their respective civilizations as much older and ancient than the broad Western culture. However, the Minoans are the oldest Western civilization and they date back to the 2000's BC while China's Xia dynasty goes back to 2000 BC and India's Harappan Civilization goes back to 3300 BC . None of these seem too far off from one another so why are Asian societies portrayed and portray themselves as so much older than the West?
While not exactly this question, I discussed some similar issues in this answer. A lot of this question in particular relates to when and how the modern Euro-American West discovered it's most ancient predecessors. The bronze age history of China has been documented as history in the same form (The Record of the Grand Historian) for most of 2000 years. The Minoans and Myceneans were functionally forgotten outside of some vague allusions in mythology after the Bronze Age.
I'd actually argue that the Harappan civilization is even less directly connected to modern India than the Minoans are to Greece. It was similarly forgotten, but has a crucial difference as well. We know so little about it. There are not many sites; there are not any identifiable references in mythology; their writing system is wholly unknown and not as well documented as Linear A. We also can't clearly identify how the Harappans connect to later Indian society. There's no direct link between the Indus Valley culture and the Vedic Aryan culture or the post-Vedic kingdoms. The Dravidian cultures of southern India are more promising, but there are no verifiable links.
Both are also tied into issues of modern nationalism. Both China and India make claims of direct descent from impossibly ancient times. Hindu Nationalists in particular tend to cite 10,000 years of continuous culture which is obviously unrealistic, and yet it's very easy to find that false claim.