That's not a fair assessment. To say the CCP has brainwashed the Chinese populous to believe Taiwan is a part of China is a gross misrepresentation of the issue. Mainland China's desire to reunite Taiwan is based on ideological grounds, in spite of practical reasons.
Practically speaking, it should be self-evident. Taiwan is to China as Cuba was to the United States. The US sells armaments to Taiwan regularly, for example, much to China's chagrin. China could view it as a US outpost in the Pacific and a key territory if, God forbid, war ever broke out between US and China. And while there aren't official relations between the US and Taiwan, the relationship is still very close. It's one of the few reasons why China isn't willing to invade Taiwan outright, because they know that the US will support Taiwan and US-backed allies (Japan, South Korea, Philippines) would nominally support Taiwan.
Ideologically speaking, however, tells a very different story. You have to look to history. Way back to the Qing Dynasty, Taiwan has been considered part of China. Google it, and you'll see the territorial extent of the last dynasty. But the Qing Dynasty also witnessed Western and Japanese encroachment of its lands, territory, and sovereignty.
For example, it suffered humiliation at the hands of the British in the two Opium Wars, as well as the seizure of Hong Kong and the forced opening of its ports to Western traders. France, Russia, Germany, Japan, and others also started divvying up the lucrative markets in China, much in the same way they did in Africa. They started to do so with "Spheres of Influence" for each Great Power. Britain, for example had exclusive trading rights in provinces such as Hunan, Jiangsu, and Guangdong. But the local populous was much more united and did have sense of national identity. So when farmers and peasants witnessed Christian missionaries and civil engineers virtually alter the landscape of their nation (which has been left untouched for generations). This left a very bitter taste in their mouth. One of the reasons why the Great Powers gave up on dividing China into formal colonies was because of the Boxer Rebellion. But that is a story for another time.
Regardless, foreign influence in China and the need to "support the [Dynasty], destroy the foreigners" started to ingrain itself in the collective psyche of the Chinese people. And when the first Sino-Japanese War hit, Japan annexed Manchuria and Taiwan in 1895. This further cemented in most Chinese people's minds a sort of xenophobic attitude towards foreigners, and the need to fight back and regain territory that was wrongfully taken from them.
World War Two didn't help ease the anxiety China has of foreigners either. The Japanese military was just as brutal to the Chinese populace as the Germans were to the Slavs in Eastern Europe or the Jews. And for many Chinese, this is still a point of contention. To regain a sense of being whole, the Chinese people want to see a reunited China. This includes Taiwan, which has been inhabited by Han Chinese as early as the 13th century, and a formal territory since the 17th.
To say that China has brainwashed its citizens into believing Taiwan is a part of China, I dare say, is ignorant. The CCP doesn't need to brainwash its citizens into believing so, because the populace believed it before the CCP even came to power. Even the ROC government's claims on China shows how rooted this desire for a united China is in all Chinese (with the older generations at the very least).
Does the CCP reinforce and perpetuate this mentality? Absolutely.
But the origins of this mentality are not the result of brainwashing and propaganda, and labeling the Chinese as mindless slaves to a propaganda machine is ultimately an unproductive narrative to push if we want to ease tensions across the strait.
