Single neurons show complex responses during cognitive tasks and dimensionality reduction has been a successful approach for finding structure in heterogeneous neural activity. When applied to neural response patterns in a population state space, where each axis represents activity of one neuron, dimensionality reduction methods often reveal low-dimensional manifolds encoding behaviorally relevant variables. Alternatively, when the same responses are viewed in a selectivity space, where each axis represents a response feature (e.g., task condition or time), dimensionality reduction methods can reveal clustering of neurons into distinct functional cell types with similar tuning properties. However, recordings from higher cortical areas indicate that neurons respond to seemingly random mixtures of task variables, so that boundaries defining functional types are not apparent. Theoretical studies have not yet provided precise predictions for when functional cell types should emerge in neural circuits and when, alternatively, neurons exhibit random mixed selectivity. To address this gap, we studied the structure of single-neuron selectivity in firing-rate recurrent neural network models. In these models, we mathematically proved that the embedding dimension of the neural manifold defines the number of functional cell types in a network, with the number of types converging to precisely n as the total variance above the n-th principal component of population activity goes to zero. This relationship is a consequence of the firing-rate non-linearity and does not hold in the synaptic activation space. We validated this theoretical prediction in task-optimized recurrent neural networks by directly controlling the embedding dimension of population dynamics during training. We further confirmed our theory in brain-wide Neuropixels recordings from mice during complex behavior. These results provide a clear characterization of the conditions under which distinct functional cell types emerge in neural circuits and show that random mixed selectivity only arises when population dynamics are high-dimensional.