I may be mistaken here, but I think I the first time I read the term 2.5D was way back when the original Klonoa was released. I always thought it was rather fitting? Then again, "3D sidescroller" might technically be a better description.
I heard of it way back when Yoshi's Story for N64 came out.
As you know, that's a sidescroller platformer peppered with some isometrics coming toward the camera to give the illusion of a full cardboard/denim world:
First Level - Yoshi's Story(notice around the 2:00 mark, when Yoshi has to groundpound near an Pach.E.Derm enemy in order to take the alternate road.
IMO, the term "2.5D" is both silly and misleading. There is no 2nd.5 dimension. There's length, height, and depth. There can be more dimensions than that according to the laws of Physics, but there cannot be 'half' of a dimension. I get that it's supposed to mean "giving the illusion of a third dimension", but if you think about it that way, there are many games since even the NES that achieve such things.
Technically, the floor effect on the old arcade/SNES Street Fighter II games is an example of that:
Street Fighter II: The World Warrior (SNES Walkthrough - HD)The floor in this dojo moves to give a pseudo-depth effect.
If you wanna go further back, even Kirby's Adventure did it with the Butter Building:
Kirby's Adventure: Butter Building 2Around the 0:15m mark you can see an effect on the building giving the illusion of depth/roundness.
A way in which things get confusing is this: Technically, you're moving in 3D in Beat 'em Ups. So is "2.5D" restricted only to platformers? After all, in a Beat 'em up, "Up" is replaced by "going in/out of the playfield' and height is represented by jumping.
What about Isometric platformers like Super Mario RPG? What are those?
This is in my opinion what makes the term "2.5D" a superfluous buzzword term, rather than an actual term. Yeah, people will kinda 'get' what you're talking about when you say it... but if you're restricting it to platformers... why not just say "Polygonal Sidescroller" (like Kirby & the Crystal Shards, Castlevania: Lords of Shadow: Mirror of Fate, or Bionic Commando: ReArmed), and say "sprite-based sidescroller" (anything made with sprites and tiles instead of polygon models)?
It's simpler and everyone will be on the same page when they hear what you're saying. The term "platformer" can be tacked on at the end of 'sidescroller' to further explain what kind of game it'll be... so you can say "polygonal sidescroller platformer" for New Super Mario Bros., "polygonal sidescroller fighter" for Super Smash Bros. "sprite-based sidescroller puzzle" for Krusty's Super Fun House or A Boy & His Blob, etc. Symphony of the Night would be a 'sprite-based sidescroller adventure', even though it is peppered with polygonal elements.
You can also use pixel-art instead of sprite-based and everyone will know what you're talking about.
Games that mess with you in that regard would be very few...
I can think of "Fez" as messing with things a lot, as it's a pixel-art sidescroller puzzle.. but its main mechanic is switching dimension perspectives (but honestly, unless you're in the middle of flipping or are on NewGame+, you're going to spend most of your time in sidescrolling mode).
Super Paper Mario is another odd animal in that it's using pixel-art visuals for the characters, but they're using polygon models for everything else. You've also got full movement in all three dimensions, but only when flipping... so really this is not always a sidescroller, but it IS a platformer, you get it?