It is not an experimentally study. It's observational, e.g. from the paper: "we exploit the abrupt end of United Kingdom sugar rationing in September 1953 as a natural experiment."

They didn't perform an experiment, they took someone else's observational data from a real-world event, and attempted to interpret it as if it had been an experiment.

OK, quasi-experimental, a discontinuity design, surely? It's not purely observational, as you've got (albeit imperfect) treatment and control, built around (presumably very similar) people born on either side of a cut-off, simulating randomisation.

Observational would be simply looking at sugar consumption across a cohort and having that as a covariate.

The fact someone else gathered the data has nothing to do with it.

Edit: It's a widely-used research technique, which attempts to address this correlation-causation issue without having to do RCTs.

https://medium.com/@arun.subram456/causal-inference-regressi...