You can destabilize the finance system by creating free money, in fact, according to exogenous money theories, it was always possible to create money out of thin air. But until cca 1980s, the creation of new money was confined to financial institutions (not just government) and heavily regulated.

The regulation was gradually lost (that's why we are having these asset bubbles and financial crises), and Bitcoin democratized the money creation even more.

But regardless whether you believe or not that Bitcoin can destabilize financial system, ultimately, if things go south, China can actually react and pull the plug on the financial industry, limit its power in the economy at large, because it owns big chunk of it (or can make sure it owns it). In the West, however, this is a political issue, and governments can only watch the financial system fail and cause another big recession (or at least, huge misallocation of resources, for example in real estate).

So why would China care? Geopolitically, they have the upper hand. We should care, though!

The Chinese government is not usually in the business of recommending specific investments. (However, they might discourage people from investing into known scams.)

The source of this claim seem to be remarks on cryptocurrency that the People's Bank of China's vice president, Li Bo, made at the Boao Forum for Asia: http://www.cs.com.cn/yh/04/202104/t20210419_6158938.html

Specifically, he said that stablecoins would have to subject themselves to the same strict regulations as banks if they want to become widely used payment channels, and that cryptocurrencies aren't per se currencies, but a different kind of investment, and that, like other countries, they're currently investigating how to regulate them to manage the associated risks.

How that turned into an "endorsement" I don't know for sure, but I suspect hype factories trying to sell bitcoin using whatever story works.