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 ${\ensuremath{\lambda}}_{C}$ only depend on the ratio $\ensuremath{\epsilon}/T$, where $\ensuremath{\epsilon}$ 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 ${\ensuremath{\lambda}}_{R}$. The scaling functions are observed to depend only on the dimensionless ratio $\ensuremath{\epsilon}/T$. 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 $\ensuremath{\epsilon}$ and $T$. Conditions limiting the validity of this ``superuniversality'' are discussed.