What differences exist between natural rain and human “rain”?
The only physically measurable difference between rain from seeded clouds and unseeded is in the seeds/ cloud condensation nuclei/ aerosols that are the surface that moisture can condensate onto and form cloud droplets.
You could collect rainwater, evaporate it and the remaining solids were the substances acting as cloud condensation nuclei. This process would get some aerosols from the ambient air, but simultaneously using a device for collecting aerosols and reducing their composition from the rainwater one would lessen this problem. After analyzing the solids one could do some guesswork whether the aerosols were natural or influenced by humans. For example if there is actually deliberate cloud seeding happening you could find traces of silver iodide. Most likely this is not the case and simply heavy pollution has a significant impact on formation of aerosols that can act as cloud condensation nuclei. In this case there are probably some compounds that can be formed only in combustion processes and finding them would mean that they played a role on the formation of the clouds. Doing this experiment in otherwise similar locations with the other having much more pollution would probably be the sanest option. Even after all this you still wouldn't be able to accurately tell whether it would've rained more or less without the pollution. If the air is clean to begin with, adding pollution and thus aerosols the cloud formation would be enhanced and probably then rain too. But if the air already has enough cloud condensation nuclei to begin with then adding aerosols might have the opposite effect; moisture would condense into many small droplets rather a few big ones that create rain.
Another possibility would be to study this using a weather model with aerosol dynamics included using measurements of the given location to verify the results. And when the model simulates the reality accurately enough you could tamper with the aerosol fields (by adding and reducing aerosols) and see how the rain is affected. I'm not aware how accurate these kind of models are currently, but aerosol dynamics can make the model easily very heavy.