Test Moderation of Standardized Variance Components

I was wondering whether it is also possible to test if there is a significant moderation of the standardized variance components.

Yes, that can be done via DeFries-Fulker regression with double-entered data. Relevant references:

Kohler, H.P., & Rodgers, J.L. (2001). DF-analyses of heritability with double-entry twin data:
Asymptotic standard errors and efficient estimation. Behavior Genetics, 31(2), 179-191.

Rodgers, J. L., & Kohler, H. P. (2005). Reformulating and simplifying the DF analysis model.
Behavior Genetics, 35(2), 211-217.

Rodgers, J. L., & McGue, M. (1994). A simple algebraic demonstration of the validity of
Defries-Fulker analysis in unselected samples with multiple kinship levels. Behavior
Genetics, 24(3), 259-262.

I will warn you: the fact that the Purcell approach estimates moderation of the raw, not standardized, variance components is one of its strengths, because moderation of the standardized components can be misleading. See the figure in the attachment. In the trait's raw metric, the nonshared-environmental component slightly increases as the moderator (SES) increases. But. because the total phenotypic variance also increases with the moderator, the standardized nonshared-environmental component actually decreases slightly with the moderator.