It is extremely common to have datasets with more than one discrete character – in fact, one could argue that multivariate discrete datasets are the cornerstone of systematics. Nowadays, the most common multivariate discrete datasets are composed of genetic/genomic data. However, the foundations of modern phylogenetic comparative biology were laid out by Hennig (1966) and the other early cladists, who worked out methods for using discrete character data to obtain phylogenetic trees that show the evolutionary history of clades.
Almost all phylogenetic reconstruction methods that use discrete characters as data make a key assumption: that each of these characters evolves independently from one another. Mathematically, one calculates the likelihood for each single character, then multiplies this likelihood (or, equivalently, adds the log-likelihood) across all characters to obtain the likelihood of the data.
The assumption of character independence is clearly not true in general. In the case of morphological characters, structures often interact with one another to determine the fitness of an individual, and it seems very likely that those structures are not independent. In fact, some times we are specifically interested in whether or not particular sets of characters evolve independently or not. Methods that assume character independence a priori are not useful for that sort of framework.
Felsenstein (1985) made a huge impact on the field of evolutionary biology with a statistical argument about species: species can not be considered independent data points because they share an evolutionary history. Species that are most closely related to one another will covary, simply due to that shared history. Nowadays, one cannot publish a paper in comparative biology without accounting directly for the non-independence of species that evolve on a tree. However, it is still very common to ignore the non-independence of characters, even when they occur together in the same organism! Surely the shared developmental history of two characters within one body commonly leads to correlations across these characters.