I think you are confused by the difference between an invention and a patent. A patent is a legal document that provides a limited duration protection to an invention in a specific jurisdiction. The same invention may be, and often is, patented in multiple jurisdictions. Having a US patent does not mean, in any way, that the invention is "American" as the inventor and the owner of the patent (which is often not the same) may be from some other country.
As for a patent "belonging to the US government", this is in general not the case for US patents. My US patents are owned by my employer. You can identify the owner of the patent by looking for who is listed as the assignee. However, if the invention was developed at a national laboratory or agency (such as Los Alamos or NASA) then it is likely the owner would be the US government. I guess you could consider those to be "American Invention", but even then it is possible that the inventor is not an American citizen.
If you insist on assigning a nationality to an invention (not a patent), then I suppose you could either refer to the inventor's nationality or perhaps the assignee's. This isn't altogether clean as there can be multiple inventors on the same patent and multinational companies can have facilities in several countries.