Colloidal building blocks with re-configurable shapes and dynamic interactions can exhibit unusual self-assembly behaviors and pathways. In this work, we consider the phase behavior of colloids coated with surface-mobile polymer brushes that behave as "dynamic surfactants." Unlike traditional polymer-grafted colloids, we show that colloids coated with dynamic surfactants can acquire anisotropic macroscopic assemblies, even for spherical colloids with isotropic attractive interactions. We use Brownian Dynamics simulations and dynamic density functional theory to demonstrate that time-dependent reorganization of the dynamic surfactants leads to phase diagrams with anisotropic assemblies. We observed that the microscopic polymer distributions impose unique geometric constraints between colloids that control their packing into lamellar, string, and vesicle phases. Our work may help discover versatile building blocks and provide extensive design freedom for assembly out of thermodynamic equilibrium.