We use perturbative renormalization group theory to study the kinetics of protein aggregation phenomena in a unified manner across multiple timescales. Using this approach, we find that, irrespective of the specific molecular details or experimental conditions, filamentous assembly systems display universal behavior in time. Moreover, we show that the universality classes for protein aggregation correspond to simple autocatalytic processes and that the diversity of behavior in these systems is determined solely by the reaction order for secondary nucleation with respect to the protein concentration, which labels all possible universality classes. We validate these predictions on experimental data for the aggregation of several different proteins at several different initial concentrations, which by appropriate coordinate transformations we are able to collapse onto universal kinetic growth curves. These results establish the power of the perturbative renormalization group in distilling the ultimately simple temporal behavior of complex protein aggregation systems, creating the possibility to study the kinetics of general self-assembly phenomena in a unified fashion.