The Security Council Does Not Determine Sovereignty
Sovereignty is the term for one state being able to govern a piece of territory. The United Nations has no process in place for determining who has sovereignty over some territory.
The General Assembly may adopt a resolution supporting a state's ability to govern a territory (which can be vetoed), but it is not entirely binding. Similarly, the International Court of Justice can be asked to make a decision regarding the sovereignty over some territory, but that requires both parties to submit to the ruling.
In this sense, it really doesn't matter that China is on the Security Council. Additionally, it is counter-balanced by the presence of the United States, who could veto any similar claim by China.
So how is sovereignty determined? The most important concept is reciprocity. A state is sovereign over some territory when other states recognize its sovereignty. For example, if other states seek permission from the United States enter the border around the island, or land there, then they are recognizing the United States' sovereignty over that land (and not China's).
Additionally, there are some other things to look for when determining who is sovereign in some territory. These are less important than reciprocity.
- The ability to create and enforce laws in that area.
- Recognition of sovereignty from the people living in that territory (for example, by paying taxes, voting in elections, serving on juries, etc.)
- A treaty wherein two states agree who has sovereignty
- A history of governing the territory, or a historic claim on that territory
If you would like a short primer on sovereignty, this document by National Unity Government seems pretty good. I'm not familiar with this organization, but the content matches what I covered in my international law coursework (in an American university in the early 2010's). You could ignore the section on Aboriginal rights.
If you are interested in a very long article about sovereignty, try the Stanford Encyclopedia of Philosophy article.