I'd say the main two for the vanilla cast are "speed" and "mobility." The first category is comprised of sonic and metal sonic, while the second has tails and knuckles.
Speed characters have practically no verticality, but can go speedyfast horizontally. Mobility characters have, ya know, good mobility and can go pretty much anywhere in a stage, but are pretty slow when doing so (though ironically they're still the fastest characters in the game because they can often just skip large parts of a level)
The nospin characters are more complicated. Movement wise, Fang feels like a mix between the two categories. He has less verticality than the mobility characters, but the tailbounce is a lot faster and burstier(?) than flying or climbing. I've also heard him described as a "figher" character, which does make sense considering most people have only ever used him for getting boss emblems.
Amy is also there. I don't know. The hammer provides such negligible upside that her most distinct feature is probably having the highest jump height in the game. I don't think amy falls into any category, I think she's just sort of a normal person.
Funny enough, speed, mobility, and fighter aligns pretty well with speed, flight, and power, but knux wouldn't be a power character. silly
As for addon characters, the speed vs mobility dichotomy sort of falls apart for many. Some characters can fit in to the categories above (inazuma's a speed character and mighty a mobility character. Kou could even be described as a fighter character), but most non-vanilla characters have both high speed and mobility. That being said, I do think addon characters can be divided into their own categories.
I'd say addon characters have 2 main categories: "vanilla-like" and "more is more."
Vanilla-like characters are characters that generally adhere to vanilla design philosophy. They'll still tend to be stronger than the vanilla cast (because basically all modded characters are), but they'll usually stick to the basic "double jump ability and spin ability" design of the vanilla characters. You're unlikely to find custom buttons being used here. Some characters in this category are shadow, gemma, or the chaotix.
Characters in the other category, "more is more" just want it all. They tend to throw vanilla design philosophy out the window and just add a whole bunch of abilities. They are very likely to use custom buttons and are often pretty OP. Some characters in this category are modern sonic or the max control characters or, at its most extreme, SMS.
It's more of a spectrum between vanilla-like and more is more than the hard binary that's laid out here, but that'll happen when trying to categorize things so whatever. Characters like elfilin or xsonic fall more in the middle of these two groups I'd say.
anyway, that's my two cents
