TLDR: They test for causality in Granger sense. It is not causality in the interventional meaning as defined in Pearl et al. (2016). If you seek for Granger causality - simply plug in any two variables. If you wish to perform causal inference - things are never so simple.
This is indeed very cool method and interesting question. However, as many other authors, Tsonis et al. (2018) call causality in Granger sense "just causality", which is in my opinion very misleading attitude. There are many definitions of causality and Granger causality is not one of them. It is something different.
To show an example how it is misleading let me first cite Tsonis et al. (2018):
"if past sea surface temperatures can be estimated from time series of sardine abundance, temperature had a measurable and recoverable influence on the population dynamics of sardines"
Ok. But what if we used something different than the temperature measures around the particular sea? What if we measured the number of sunburns got by the population of people sunbathing by the sea? Let me paraphrase their sentence:
"if past numbers of sunburns can be estimated from time series of sardine abundance, sunburns had a measurable and recoverable influence on the population dynamics of sardines"
Now, this looks bad. But it is how causality in Granger sense works. It seeks for predictors, and has purely observational meaning. If we observe unnaturally high number of sunburns this summer, will we observe changes in sardines population in the future? Very likely. But if we ban suntan cream from use, will population of sardines change? Highly unlikely.
It is important to note, that the variables Tsonis et al. (2018) say that they influence something, are arguably "very exogenous". It is extremely unlikely to find variables that influence temperature in short term, it is much more unlikely to find variables that influence cosmic radiation.
Such situation helps causal inference (and make this article even more misleading, because it makes sense) but it it is an external knowledge (assumption about DGP), which can not be derived directly from the data. And causal inference always requires such external knowledge.
Pearl, J., Glymour, M. and Jewell, N.P., 2016. Causal inference in statistics: A primer. John Wiley & Sons.
Tsonis, Anastasios A., Ethan R. Deyle, Hao Ye, and George Sugihara. Convergent cross mapping: theory and an example. In Advances in nonlinear geosciences, pp. 587-600. Springer, Cham, 2018.