I think there are six different characteristics of stages. First, there's orientation of the stage, whether it's horizontally or vertically oriented. Second, there's blast radius, or how far the blast lines are from center stage. Third, the size, or how the area of the surfaces upon which characters can land. Fourth, the edge shape. Fifth, the platform heights. Finally, we must consider the stupid random stuff which comes with every stage.
Horizontal vs. Vertical:
A horizontal stage is one where the floor (the bottom most platform which cannot be fallen through) is very wide whereas the platforms do not reach proportionately high. FD is the perfect example of a horizontal stage because it doesn't have platforms. It's just the floor going from one end to the other. Perhaps the most vertical is Kongo Jungle 64 because of it's super-high platforms. Let's just leave Rainbow Cruise out of this for now. Here are the neutrals from horizontal to vertical: FD, then Dreamland, Battlefield, Yoshi's, and FoD. CPs: PS (normal, skipping the transformations), then Corneria, Brinstar, and KJ. I'm skipping Rainbow Cruise because it's a little funky. I really can't remember the CPs and am too lazy to look them up. This is important to understand because some characters like to have more aerial involvement than others. Roy, for example, hates leaving the ground. His aerials stink. As for Jiggs, she wants to be able to fly all over everywhere and be able to renew her five jumps on whatever platform wherever she likes. Roy wants a horizontal stage where Jiggs wants a vertical one. These are the two extremes (except maybe Falco might be a better example of horizontal favoritism, but you said no spacies) and everyone else falls somewhere in between based on the matchup. Link has to adapt to what type of camping will hurt the other player the most and pick orientation accordingly.
Blast Radii:
In smash, you die when you cross the blast lines. Therefore, if you want everyone to live longer, widen the blast lines. If you want deaths quicker, tighten the blast lines. Here, we introduce survivability. Link is heavy so he doesn't get hit as far as most of the other characters, has a long, varied recovery, and has an average falling speed so he neither falls too quickly to recover nor gets hit too high to come back down. In other words, Link can live for a very, very long time. Who can't live for a long time is Falco. He's a fast faller, so if you get him off the edge low, he's sunk. Therefore, Falco is very gimpable where Link is not so much. Having wide blast lines means that people with good survivability will be able to live for longer, but people with lousy survivability won't be as able to live longer because too far is too far on any stage. Therefore, if you're shooting for survivability, go for stages with wide blast radii. If you're looking for early kills, go for smaller stages. Link wants to take YL to a small stage so his superior range and power will kick YL off before the little brat can do much camping damage, so he's a bit of an exception.
Size:
Different stages have different sizes. This can be because they are wide or because they are tall or because they are both. From larges to smallest: Dreamland, FD, Battlefield, FoD, Yoshi's. Please note that stage size is independent from blast radius, thought the two usually correlate. FoD is a small stage with a large blast radius. Characters with close quarters attacks with a long reach such as Ganondorf or Marth want a small stage whereas characters with better mobility or campability want bigger stages. Link is a campy little so-and-so against the dorf, so he wants a huge stage so he can run away and rang. Against YL, who can move much more quickly than Link and can spam projectiles better than Link can, Link wants to go to a tiny stage like Yoshi's so his superior reach and power can give him an edge. Falcon and Fox have great mobility, so they're both aiming for big stages, even though they can't camp with projectiles. In stead, they camp by running in and out of your zones waiting for you to throw out the wrong attack so they can run in and punish.
Platform Height:
Low platforms mean wavelands. High platforms mean aerials. Ganondorf relies on wavelanding for mobility, so he wants fairly low platforms like on PS. Link likes his nair, so he wants high platforms like on Dreamland. Platform height from tall to short: DL, BF, FoD, Yoshi's, FD. Marth and Roy can slide all over everywhere and their f-smashes and u-tilts can go right through most platforms.
Edges:
Having no edges means that teather recoveries don't work and that edgeteching is impossible. BF has teatherable and techable edges, but both recovery techniques are more difficult.
Random Shiz:
- Samus can perfectly SH missile cancel smash missiles on PS
- M2 can spin you underneath the overhang on BF
- Randall
- Marth and Roy's f-smashes don't reach through DL's platforms, but their u-tilts do
- Teh bukkit on KJ64
- There are some platforms on Rainbow Cruise which you can fall through and others you can't. KNOW WHICH ONES ARE WHICH
- Samus has an unlimited vertical recovery on FoD
Therefore, different characters with different movesets and metagames will like different stages against different people for different reasons. See Skler's guide for specifics on individual CPs.
TL;DR: Yes, yes CPs are different for different characters.
Also, how does Marth=Falco and since when do birds like pies?
Edit: Post # 40! Yay!