Recent studies have highlighted interesting structural properties of empirical cultural states. Such a state is a collection of vectors of cultural traits of real individuals, based on which one defines a matrix of similarities between individuals. This study provides further insights about the structure encoded in these states, using concepts from random matrix theory. For generating random matrices that are appropriate as a structureless reference, we propose a null model that enforces, on average, the empirical occurrence frequency of each possible trait. With respect to this null model, the empirical similarity matrices show deviating eigenvalues, which may be signatures of cultural groups that might not be recognizable by other means. However, they can conceivably also be artifacts of arbitrary, dataset-dependent correlations between cultural variables. In order to understand this possibility, independently of any empirical information, we study a toy model which explicitly enforces a specified level of correlation in a minimally-biased way, in the simplest conceivable setting. In parallel, a second toy model is used to explicitly enforce group structure, in a very similar setting. By analyzing and comparing cultural states generated with these toy models, we show that a deviating eigenvalue, such as those observed for empirical data, can also be induced by correlations alone. Such a "false" group mode can still be distinguished from a "true" one, by evaluating the uniformity of the entries of the respective eigenvector, while checking whether this uniformity is statistically compatible with the null model. For empirical data, the eigenvector uniformities of all deviating eigenvalues are shown to be compatible with the null model, suggesting that the apparent group structure is not genuine, although a decisive statement requires further research.