The phase-ordering kinetics of the ferromagnetic two-dimensional Ising model with uniform bond disorder is investigated by intensive Monte Carlo simulations. Simple aging behavior is observed in the single-time correlator and the two-time responses and correlators. The dynamical exponent z and the autocorrelation exponent lambda(C) only depend on the ratio epsilon/T, where epsilon describes the width of the distribution of the disorder, whereas a more complicated behavior is found for the nonequilibrium exponent a of the two-time response as well as for the autoresponse exponent lambda(R). The scaling functions are observed to depend only on the dimensionless ratio epsilon/T. If the length scales are measured in terms of the time-dependent domain size L(t), the form of the scaling functions is in general independent of both epsilon and T. Conditions limiting the validity of this "superuniversality" are discussed.