The impression and definition come from anthropology and history. In this case Chinese means 中华 and there's the Chinese language, race, and culture or 华文, 汉族, 中华文化. I apologise if I don't grasp some nuance from the words in Chinese I chose but from an objective, historical point of view, these terms are consistent. The lineage and changes (or lack of changes) of these has been established with varying levels of definitiveness.
The most definitive is the ancestry Han. DNA tracing has established the genetic continuity of a Han race going back thousands of years, with an extremely homogeneous genetic profile between members of the race. 95% of the population of China and Taiwan belong to this race, with a 5% of minorities.
Next most definitive is the language. The modern form of what is called regular script (I am unsure if what it is called in Chinese but it is traditional Chinese today), was the predominant form since about 250 AD. Certain words have become archaic and literary allusions might be lost. And aside from the changes in word order, you could read texts from this period onwards (with some difficulty). The governments of China all used this script, with certain governments also using additional languages like Manchu. The way it was spoken did vary a lot even till today, but there definitely is the Chinese language that both Taiwan and China uses. The fact that people from these countries can converse in their native tongue is basic empirical proof you speak the same language.
Culture is the least definitive concept, but it's very plain you have a shared culture. It's not exactly the same (and culture varies within a country too) but historically it is very clear it comes from the same culture. Frankly it's quite a basic fact that the people on Taiwan and China have historically been part of same people.
I'd like to emphasize what the CCP says does not change anything about the history and anthropology of it. I'd I'm being blunt, both what might be called the "CCP Han identity" and "Taiwanese identity" are political movements. In the case of the CCP to associate themselves with the history of China (which has swung wildly given their behavior especially during the cultural revolution) and in the case of Taiwanese identity (which has intensified in recent decades because of the geopolitical situation).