To answer the question in your title: No. This is not a normal amount of soapsuds to see in a dishwasher after it is finished.
If there is this much soap in the washer at the end, then there is a lot of soap left on the dishes. Even though you can't taste it. Try dipping your finished dishes in a pan of clean water to see how much soap rinses off.
I have had this much suds left in a clothes washer, and it turned out I was using the wrong detergent. Check carefully that you are using the detergent specified by the manufacturer.
I hope that you have just used the wrong, or too much, detergent, because that would be such an easy fix.
The most likely cause of your soapsuds problem is not enough water. For some reason, your dishwasher is not filling completely. You can tell by opening the door (which pauses the washing action) during a wash or rinse phase and looking at the water level.
In order to really test this properly, you need to know what the normal water level looks like. I don't know exactly how you would find this out. But most dishwashers have a float valve that shuts off the water at the proper level. If you can find this switch, you can at least tell whether it is being switched by the water level.
Another thing look at is the amount of residual water remaining in the machine between cycles. If the dishwasher is not emptying completely between phases then you are rinsing with soapy water. You have to lift the debris filter to check the low water level between phases.
If you find that the dishwasher isn't filling or emptying properly, then you're looking at a plumbing problem.
Check for debris in the supply line. Check for a clogged input filter. Check for a broken or partially operating fill valve.
Check for a kinked or blocked drain line. Check for a broken or partially operating drain valve. Check for debris blocking the drain opening inside the washer under that large round screen.
Keep lots of towels handy. Wear work clothing. You will probably get splashed in the face at least once.