Nope. There’re tons of Chinese startups already with or without YC. Would US startups be required to register as Chinese companies if they receive Chinese VC money?
The US is open to arbitrary FDI so it doesn't matter. China's laws are more...complicated. For example, you can't give stock options to non-Chinese employees if your company is Chinese and not already listed, see https://www.chinalawblog.com/2018/03/china-scam-week-part-4-....
I'm not aware of any foreign funded startups in China currently. I mean, there must be some small players with private foreign funding, but it is so easy to get Chinese funding I'm not sure why anyone would even bother.
I'm betting that they will be using Chinese capital for Chinese start-ups, so there won't be any weird foreign registration requirement. It makes more sense because China still has a lot of money sloshing around looking for something to do, and YC China will look appealing in that regard, so no FDI will be needed.
But that is just a guess by an amateur, we have to wait to see how this turns out.