Ecological relationships between bacteria mediate the services that gut microbiomes provide to their hosts. Knowing the overall direction and strength of these relationships is essential to learn how ecology scales up to affect microbiome assembly, dynamics, and host health. However, whether bacterial relationships are generalizable across hosts or personalized to individual hosts is debated. Here, we apply a robust, multinomial logistic-normal modeling framework to extensive time series data (5534 samples from 56 baboon hosts over 13 years) to infer thousands of correlations in bacterial abundance in individual baboons and test the degree to which bacterial abundance correlations are ‘universal’. We also compare these patterns to two human data sets. We find that, most bacterial correlations are weak, negative, and universal across hosts, such that shared correlation patterns dominate over host-specific correlations by almost twofold. Further, taxon pairs that had inconsistent correlation signs (either positive or negative) in different hosts always had weak correlations within hosts. From the host perspective, host pairs with the most similar bacterial correlation patterns also had similar microbiome taxonomic compositions and tended to be genetic relatives. Compared to humans, universality in baboons was similar to that in human infants, and stronger than one data set from human adults. Bacterial families that showed universal correlations in human infants were often universal in baboons. Together, our work contributes new tools for analyzing the universality of bacterial associations across hosts, with implications for microbiome personalization, community assembly, and stability, and for designing microbiome interventions to improve host health.