Welcome, looking at you picture of the P-Trap With 2 x washing machine type waste pipe spigot connections.
I would first investigate if there is a fat build up in the trap, does the sink plug / drain smell ? maybe worth unscrewing P-trap from bottom of sink and see if there is any signs of blockage etc.
Then if possible, disconnect dishwasher drain hose from trap waste spigot and place the end in a bucket at ground level and see if the dishwasher then drains when operated (short cycle).
It is tricky to diagnose, but it does look as though the trap ‘main’ drain hose is rather high and may need lowering to aid drainage and also check to see if the Dishwasher hose has one or many high points (loop), it should only have one in practice as drainage water could be feeding back into the dishwasher from the water stored in dishwasher drainage hose.
Washing machines drain pumps can be stronger than dishwasher pump, so may explain why no issues with washing machine.
Pic courtesy of homeinspectiongeeks.com
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