Edit:
Digging deeper, it looks like Nature treats the Chinese Academy of Sciences (the Chinese version of US DoE National Labs) as a single institution.
I'd be curious how the US stacks up if National Lab affiliation was included in the same manner. Lots of R&D done by National Lab researchers are affiliated only with their Lab, not a university.
Also a lot of naturalized American academics can also get Chinese Academy of Sciences affiliation due to the murkiness of Chinese nationality renunciation, so that might muddy it further (eg. If I'm an academic at Harvard University with an affiliation to the CAS, which relationship does the Nature Index treat as the primary?)
Edit 2: from the Nature Index itself
"For authors who are affiliated with more than one institution, the author’s Share is split equally between each institution. The total Share for an institution is calculated by summing the Share for individual affiliated authors."
So, if a researcher at Harvard or MIT or Cal is also cross affiliated with the CAS or Tsinghua, their research output would be split equally, even if the relationship is an adjunct one with one or the other.
This sounds like the same kind of statistical gaming NUS and NTU in Singapore and KAUST in Saudi Arabia are famous for to climb up global university rankings.