You're right that this isn't a meta-study. It's a deep survey and references many similar studies, though. It certainly doesn't exist in a vacuum.

"In this section, we explore in a more direct manner the relationship between unions and income inequality, joining an extensive empirical literature examining how unions shape the income distribution."

You're right to point out that "correlation is not causation," but the study specifically addresses your concern and presents a strong argument for causation. It's not as if science can never demonstrate anything using correlation and statistical techniques, you know?

https://xkcd.com/552/
"Correlation doesn’t imply causation, but it does waggle its eyebrows suggestively and gesture furtively while mouthing ‘look over there.'"

> It notes that there is a correlation between union membership and inequality

Specifically, it notes a robust INVERSE correlation that:

* Increased union membership correlates to decreased inequality

* Decreased union membership correlates to increased inequality

> Which is interesting but not that powerful - correlation is not causation

The authors acknowledge the statistical nature of correlation, and they addressed it (they say so right in the abstract). They used the following techniques to establish a causal relationship between union density and income inequality.

* distributional decompositions

* time series regressions

* state-year regressions

* instrumental-variable strategy based on historical events like the 1935 legalization of unions and the World War II–era War Labor Board

> It might be that both trends are being driven by the financialisation of the US economy.

The authors find that policy changes which significantly reduced the cost of union organizing (e.g., the Wagner Act and the National War Labor Board during WWII) led to lasting increases in state-level union density and corresponding reductions in income inequality. These effects were specific to the periods when these policies were active and had no similar impact in other times, such as during the Korean War, which did not explicitly promote union organization. This is a cause very different from a wild guess at "financialisation of the US economy."

Furthermore, the study highlights that unions were particularly effective in reducing inequality by increasing the wages of less-educated and nonwhite workers. During periods of high union density, the wage gap between union and nonunion workers was substantial, contributing significantly to overall income equality.

> It finds that union households earn a premium over non-union households. Again, because of the nature of the study that doesn't tell us much about the impacts of unions.

The study does more than just observe a premium; it provides historical and statistical context to argue that the premium is associated with union activities, such as collective bargaining. The consistent premium over many decades, despite changes in union density, suggests a link between union presence and wage levels.

In short, no, neither union membership or inequality are evidenced as show "both caused by financialization of the economy." They the latter correlated to the former, and was tested for causation. The former's uptake can include economic considerations, but it also correlated directly to governance policy directly targeting labor law.