I'm not sure what you mean by simultaneously processing the data from two different PRFs, but what the example does is to process the data that corresponds to the same range, with the assumption that no matter which PRF is used, the clutter does not change from pulse to pulse.
This approach does not work with range Doppler response because the processing in range Doppler response requires a fixed PRF so the Doppler can be extracted. In that case you actually want to get the Doppler information. On the other hand, staggered PRF is often related to MTI, where it only intends to know whether there are targets in motion. The value of Doppler for those moving targets are not sought.