I don't think China has grudge against Wikimedia specifically, the official policy of theirs has always been that if you recognize Taiwan, you are no friend of them. I don't think the Chinese delegate there cares about free knowledge, further it's their job literally to say no to Taiwan recongition anywhere on earth. So why should we be surprised that China does this when the United States federal government don't ca…
First of all, no private organisation can "recognize" a state. That is a privilege for states. Private organisations can only boycott (non-)states by not doing any business with said (non-)citizens. Why in the world Wikimedia do that, and how? Second, this policy by china is the problem. Not the lack of obedience by the rest of the world.
Yes they can. A private organization could just treat it like a state, in the same way that it treats any other state.
The consequences of that depend on what the organization does.