Would you say that is a type of S-trap?
Yes… and, where’s the vent?
I would call it an s-trap with an unvented p-trap. It seems to me that this setup is much worse than the stand alone s-trap since as one trap drains the other trap would be sucked dry.
Does Kansas allow the discharge from the dishwasher to be attached to the garbage disposal?
Even if they do, not like the one that is pictured.
It is allowed here
Dual sinks should not be double trapped either. The water flowing past the lower trap assmbly can siphon the water plug from the lower trap and allow for sewer gases to flow back into the living space. I would recommend a complete reconfiguration of the drain lines in that area.
Nice high loop on the dishwasher drain.
What’s even worse is that this looks like new construction.