Pics from Kirby GC, The new Zelda, and Mario Baseball. Yeah, Mario Baseball.

[A Black Falcon]

Kirby 64 uses all three of its dimensions. The math is still 3D. Just because you don't move along the z axis doesn't make it 2D. There are still three visible dimensions in the game.

Yes, there are three visible dimensions. But the actual gameplay is just 2d that weaves around in this 3d environment... it probably would be harder to do than a standard 2d game, and it makes it look more unique (with curving and branching paths that split from eachother and stuff), but calling it almost a different genre? I'd definitely not say that. It's just a somewhat different take on the same genre.

Hmm... that is a good question, about how you would go about designing it. I guess you would just make the level and then later set the camera and path (like for GGA where it splits and the different paths seperate into totally different areas)... so from a technical perspective yes it is 3d. But from a gameplay perspective it isn't really. Like take Star Soldier: Vanishing Earth (the one topdown scrolling ship-shooter for the n64) for the N64. It's "3d" as in it uses polygons, ships fly down into the field, guns below the area you can shoot shoot up at you (and you can't hit them), etc, but all you can do is go along on this plane... obviously it's using a 3d engine as there are all the bullets and ships that use the third dimension, but for you yourself it might as well not be there (and those things I described aren't things that would be impossible in 2d either, of course...).

The way you differentiate between 2D games and 3D games is by looking at the math. Any 2D game can kinda look 3D (like Doom), but Doom's engine does 2D math while Mario 64's does 3D math, which tells the graphics hardware to do 2D or 3D things, respectively.

It gets fuzzy for cases like Duke Nukem 3D, though... that is more 3d than non but it still is using some of those 2d tricks.