How does color LCD work on the Pokémon Pikachu 2 GS?
You're right that it's not the same as the technology used on (modern) laptops or flat screen displays.  It's almost undoubtedly a color passive matrix display.  Nintendo used that kind of display on the Color Gameboy as well.  For small displays that don't have to act quickly, a color passive matrix display is a good fit.  It's much cheaper to make than an active display which is why this technology, once common on laptop computers, was attractive to manufacturers.  

If you look at the screen very closely, you can see the difference:


Instead of having individual display elements for discrete red, green, and blue pixels, the matrix works by having three layers of liquid crystals each separated  by  a  color  filter -- essentially three stacked monochrome LCD arrays. By turning on one or more of the pixels in each layer, we get color. 

How do stacked color displays work?

There are a number of descriptions of these available, but many are 30 or more years old and may not be easy to find.  One that can be found online is US patent 4917465 which has good illustrations and shows a particular system using yellow, cyan and magenta filters for a backlit display, but the background section of the patent gives a good description of the different approaches to color LCD before 1990 and the principles are the same as used in reflective displays such as the one discussed here.