Well, at least the GDP identity is an accounting identity, so it has some value, but the causal relationships are not there.
Every mathematical statement is a tautology (in that they are inevitable conclusions logically derived from a set of axioms). If the axioms constitute a decent abstraction of reality, then you can try to test causal relationships uncovered by logic, or use them in practical ways (Euclidean geometry, Newton's laws etc).
You can read Kaya's identity as per capita CO2 emissions from human sources is equal to per capita GDP times energy use per GDP unit times CO2 emissions per energy unit production. So, you have three ways of reducing per capita CO2 emissions if you are a global central planner (pushing aside the miserable implications of having a global central planner): 1) reduce living standards; 2) use energy more efficiently; and 3) generate less CO2 when producing energy.
And, if all you care about is reducing total CO2 emissions, start culling people or reduce the birth rate. This appeals to the 1970s neo-Malthusian, "the world is going to run out of food" crowd cause they make a living selling catastrophe. Combined with neo-Marxists who like to emphasize distribution much more than the absolute level of wealth, they like to ignore how living standards have improved[1] contemporaneously with increased CO2 emissions.
[1]: https://www.vox.com/the-big-idea/2016/12/23/14062168/history...