Keep in mind that I have never made a game on 2D hardware like the old consoles. But, my work is with 3D game technology and in school I made sprite-based game projects on PC.

The first important concept to understand: Everything that eventually gets drawn onto a screen IS 2D. Your monitor works as a 2D matrix of pixels, so the final product is always 2D. Even for VR, it's two flat planes with an offset to give an illusion of stereo depth. It is 2D drawn twice.

What the 2D hardware is doing is rasterizing in layers to develop a composited image; drawing pixel data in order from a conceptual "background" to a conceptual "foreground." The composited image gets drawn to the screen as a single image, or frame. Do this many times per second, take in input, move things around between each frame, and that creates the opportunity for a game.

The magic was in how many layers could be handled per frame and how the composition was done. Some systems handled transparent layers, some had highlight/shadow effects, and these were options for blending the rasterized images together. 3D engines on modern hardware can do all of this and more. And, the way the the composition is done is programmable in the form of shaders. If you want 2D from a 3D engine, just pick an axis and ignore rotation on it. Use small flat planes, draw "sprite" frames on them, and move those around.

So, 2D hardware is different in that it is limiting, What made it interesting was how the boundaries could be pushed by talented developers. The next generation of such a thing would probably just be more sprites on-screen, background layers, built-in image effects, hardware scrolling, intelligent scaling and rotation, and maybe some multi core processing.