I should clarify, this is fair. When I say "Chinese culture," I mean in the Epic sense - five thousand years of Chinese heritage and cultural tradition, mushed together from all the provinces and colonies. It's not a very well defined thing, so your comment is quite fair. It includes things like the use of the Han writing system, a similar linguistic heritage, certain medical beliefs, a smorgasbord of culinary traditions, a flavor of bureaucracy, etc.

With this in mind, the PRC has slowly been eroding many aspects of this "traditional" Chinese culture, such as the high respect paid towards intellectuals, teachers, and philosophers (through the use of reeducation camps and propaganda). Meanwhile, Taiwan starts to discover new things about the natures of traditional Chinese culture, like its compatibility with quite liberal concepts (same-sex marriage, as an example).

Many Taiwanese people will say their cultural identity aligns with the thousands of years of Chinese history, while rejecting the last 60 years or so of change on the mainland.

Nowadays someone might say "Chinese culture" and be taken to mean things like "tendency towards tyrannical capitalism masked as communism" or "tendency to imprison people for thought-crimes," so I appreciate you bringing this up.