Functional redundancy has been hypothesised to be at the core of the well-evidenced relation between high ecological microbiome diversity and human health. Here, we conceptualise and operationalise functional redundancy on a single-trait level for functionally annotated microbial communities, utilising an information-theoretic approach based on relative entropy that also allows for the quantification of functional interdependency across species. Via constraint-based microbiome community modelling of a public faecal metagenomic dataset, we demonstrate that the strength of the relation between species diversity and functional redundancy is dependent on specific attributes of the function under consideration such as the rarity and the occurring functional interdependencies. Moreover, by integrating faecal metabolome data, we highlight that measures of functional redundancy have correlates in the host's metabolome. We further demonstrate that microbiomes sampled from colorectal cancer patients display higher levels of species-species functional interdependencies than those of healthy controls. By analysing microbiome community models from an inflammatory bowel disease (IBD) study, we show that although species diversity decreased in IBD subjects, functional redundancy increased for certain metabolites, notably hydrogen sulphide. This finding highlights their potential to provide valuable insights beyond species diversity. Here, we formalise the concept of functional redundancy in microbial communities and demonstrate its usefulness in real microbiome data, providing a foundation for a deeper understanding of how microbiome diversity shapes the functional capacities of a microbiome.