Natosha, your cabinets are beautiful, and some of the best made.
According to the StarMark website, the end panels are 3/4" thick veneer plywood, and are set in 1/4" from the ends of the faceframe, unless flush end panels are specified, which eliminates this reveal.
It's impossible to tell from the picture if a sink cabinet or a sink front is used in this installation. If it is a front, there is a 5" return at each side. Regardless, there should be sufficient material to fasten the dishwasher to, filling in with 1/4" plywood if necessary to accommodate the reveal. I also can not tell if there is a filler to the left of the machine. I would think one might be necessary to assure the DW door opens without scraping the wall.
As the DW is designed to have a cabinet panel attached, I can't imagine Whirlpool or any other manufacturer having one that doesn't line up with adjoining cabinetry, as that is the whole point. The common thickness of the wood panels is 3/4", but may vary by door style. The specifications of the DW show insulation on the back, but say it can be compressed. The measurement of the machine back to front, not including the decorative panel, is 24". This is the same as the depth of standard base cabinets. It's possible something is obstructing it from going back into the opening fully - perhaps bunched-up insulation, debris, or wiring or piping out of place. I would remove the machine and check for obstructions.
I hope you find a solution, so you can fully enjoy the new kitchen.