Sprite sizes for 8bit and 16bit consoles had to be some number of hardware tiles in width of tiles. Note that these weren't the same as world/graphical tiles, hardware tiles were 8x8 on most of them, while most games used a graphical tile size of 16x16, so each tile was actually 2x2 hardware tiles. The sprite didn't have to fill up the entirety of its tiles, but artists usually made their work fill up as much space as possible to get more detail into the art.
Beyond working with specific files, they could be any size that could be rendered quickly enough and looked good.
For example, the NES could render 8 sprite tiles per scanline, and had a memory limit of 64 sprite tiles, this is why NES sprites tended to be very small and you rarely had very hectic, sprite-heavy scenes. If you made a 4x4 tile character (32x32 pixels, 16 tiles total), you already used up a quarter of your entire sprite budget for that frame, and half of your per-scanline sprite budget for 32 of your scanlines.
The SNES had higher limits for the number of sprites (and therefore sprite tiles) in memory and the number of tiles per scanline.
Beyond being made up of 8x8 tiles, there were also limits on total sprite size, though I'm not familiar with the specifics. However, I am pretty sure there were no limits on the sprite proportions, at least within the size limit. Mario was 2x3 tiles (16x24 pixels), for example.
Unless you're aiming specifically to follow some specific hardware limitations (in which case, look up the specifics), don't worry about the proportions. You can use multiples of 8 as a guide or to help get the feel you want, but even that isn't really necessary. You may want to limit yourself to 64x64px total, as that tended to be the limit in 16-bit consoles. They created the illusion of larger sprites by using the background layer and combining multiple independent sprites when needed, but mostly they stuck to smaller sprites for various reasons.
Beyond the hardware limit, there was also aesthetics and clarity to keep in mind - a very small sprite would be easily lost on the comparatively large screen (e.g. a 256x224px SNES screen), while a large sprite wouldn't leave much room for other information. Sprites that were 16px, 24px, or 32px tall were a pretty good size on the screen - easy enough to see, but left enough room for other sprites, a large chunk of the world, the HUD, etc. Larger sprites quickly "fill up" the screen, and even if other hardware limits weren't a problem, this would make the game unpleasant to play because it would feel crowded. For this reason, sprites were kept small, and only setpieces such as bosses were large.