No. The lower plate is not moving.Not about momentum. But about the description of the two plates. Here is what you said:
"To get an idea of viscous friction, think extreme cases like very viscous fluids like molasses, pancake syrup, and corn syrup. Imagine that you have the fluid contained between two horizontal parallel plates, and you are trying to slide the top plate to the right at constant speed. You need to exert a force on the upper plate to the right and a force on the lower plate to the left to hold it in place. OK so far?"
This is how I interpreted it:
View attachment 235821
It seems you are wanting to oppose the fluid moving with the plates - so "holding it in place" as you said ("it" being the fluid) while you move the plates, the upper plate to the right and the lower plate to the left.