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The arise of disagreement is an emergent phenomenon that can be observed within a growing social group and, beyond a certain threshold, can lead to group fragmentation. To better understand how disagreement emerges, we introduce an analytically tractable model of group formation where individuals have multidimensional binary opinions and the group grows through a noisy homophily principle, i.e., like-minded individuals attract each other with exceptions occurring with some small probability. Assuming that the level of disagreement is correlated with the number of different opinions coexisting within the group, we find analytically and numerically that in growing groups disagreement emerges spontaneously regardless of how small the noise in the system is. Moreover, for groups of infinite size, fragmentation is inevitable. We also show that the model outcomes are robust under different group growth mechanisms.