Ground collisions in YI are most definitely tile-based.
HOWEVER, it is not a simple binary system where tiles are either solid or not - instead they have a wide variety of different tile types.
For example, Mega Man X has different tile types to represent different sloped tiles - when the player collides with a slope tile, the game can determine from the tile type and the player's x-coordinate what the actual ground height is at that location.
Also note that most such games do not do full bounding-box collisions with the ground, but instead use a small number of single-pixel "sensors", because it's faster to test single points against a tile map rather than whole bounding boxes.
Also, consoles of that time did a lot of the work with hardware, so probably the detection was made by the hardware checking if sprites overlaped. I know the NES did it that way, so I guess the SNES did something similar.
Actually collision detection on both NES and SNES was done entirely via software.
The NES' "hardware collision detection", such as it is, is strictly limited to determining if sprite index 0 is overlapping non-transparent background pixels - it's only real purpose is as a timing marker for raster effects.