Only thing I see wrong with that picture is the fact the trap on the disposal is having the extension between the U bend and the quarter bend, and the dishwasher does not loop high enough.
To fix the height problem you need to open the wall and lower the sanitary tee in the wall. Having two p-traps is perfectly legal and having them on a single vent is legal as well. Here in Illinois the code states they want the disposal on its own trap separate from the other side of the sink, and you can not use the knock out on the disposal for a drain point for the dishwasher.
Section 890.710 Food Waste Disposal Units
a) Installation. Food waste disposal units shall be trapped separately from any other fixture or compartment, shall be connected directly to the sanitary drainage system, and shall be properly vented. Dishwashers shall not discharge into food waste disposal units. Units may have either automatic or hand-operated water supply control. (See Section 890.1130(a), (b) and (c).) [
b) Commercial-Type Grinders. Commercial-type food grinders shall be provided with a waste line at least 2 inches in diameter. (See Appendix F: Illustration D.)
Section 890.770 Dishwashing Machines
a) Domestic Dishwasher (Private Residence). When a domestic dishwashing machine drain line is connected to the house side of a trap from a sink, the drain from the dishwasher shall be carried up to the underside of the spill rim of the sink. Dishwashing machines shall discharge separately into a trap or tail piece of the kitchen sink and shall not connect to the food waste disposal unit.
b) Water Supply Connection. The water supply to commercial dishwashing machines shall be connected through an air gap or by means of proper backflow protection, e.g., a non-pressure type (atmospheric) vacuum breaker or a dual check valve backflow preventer assembly (DuC), depending upon the circumstances.
c) Commercial dishwashing machines shall indirectly discharge to a proper receptor connected to the drainage system or as permitted in Section 890.1010(a).
d) Hot Water. A commercial dishwashing machine or similar dishwashing equipment that relies upon hot water for sanitizing dishes and utensils, rather than chemicals for sanitizing, shall provide rinse water at 180 degrees F., except for a single-tank, stationary-rack, single temperature dishwashing machine which shall provide a rinse water temperature of 165 degrees F., in accordance with Section 750.830(h) of the "Food Service Sanitation Code" (77 Ill. Adm. Code 750).
Now if you have a look at tis picture from the Illinois code book, they show a double wye with two traps and a c.o in the middle. So it is a legal.