Longitudinal microbiome studies offer critical insights into microbial community dynamics, helping to distinguish true biological signals from interindividual variability. Tensor decompositions, such as CANDECOMP/PARAFAC (CP), have been applied to analyze longitudinal microbiome data by arranging temporal measurements as a third-order tensor with modes representing taxa, time, and hosts. While these methods have proven useful in revealing the underlying structures in such data, they are limited in their ability to capture host-specific microbial dynamics including individual accelerated or delayed phenomena. To address this limitation, we use the PARAFAC2 model, a more flexible tensor model, which can account for host-specific differences in temporal trajectories of microbial communities. We analyze longitudinal microbiome data from the COPSAC2010 (Copenhagen Prospective Studies on Asthma in Childhood) cohort, tracking gut microbiome maturation in children over their first six years of life, along with data from the FARMM (Food and Resulting Microbial Metabolites) study, examining dietary effects before and after microbiota depletion. We show that both CP and PARAFAC2 decompositions reveal meaningful microbial signatures, including compositional shifts associated with birth mode, presence of older siblings, and dietary interventions. However, while CP captures the main microbial trends in time, PARAFAC2 uncovers host- and subgroup-specific developmental trajectories, offering a more nuanced view of microbiome maturation, highlighting its potential to enhance longitudinal microbiome data analysis. In addition, we discuss the interpretability of the extracted patterns facilitated by the uniqueness properties of CP and PARAFAC2, and discuss potential challenges related to the generalization of the patterns through the concept of replicability.