I don't know if it's only my impression or not, but Jazz seems to be not especially useful for multiplayer mods, due to special moves. Jumping up - well, seems rather like a giving good opportunity to rival to be shot (nowhere near as useful as double jump for Spazz). And no "horizontal" kick, since it's swapped with the high jump. Even Lori seems to have more optimal set of special moves.

Yeah, I know that all 3 characters are doing the basic stuff (shooting, running and jumping) in the same way and those are the crucial for almost any MP mode, but when it comes to differences - is Jazz really the worst choice?

In multiplayer, the vast majority of “serious” players use Spaz for his superior double jump. His jump is the highest and the most versatile for moving around the map. Speed is at least half the effort of doing well in multiplayer.

Jazz would be somewhat tied with Lori as the worst character in multiplayer, but it’s still situation dependent. Special moves aren’t really convenient in traditional multiplayer modes unless they aid you in moving around the level. Jazz’s uppercut gives him some height, while Lori has no means for that through special moves (no uppercut or double jump).

Keep in mind that most modern Jazz2 servers utilize JJ2+ which adds support for mutators, among other things. Two common mutators are uppercut.mut and the more experimental charbalance.mut, or whatever it was called. These try to balance out the special moves for Jazz and Lori in order to better match them with Spaz.

Out of curiosity - are those mutators (or any mutators) by default active in JJ2+?

I admit that I'm not an expert in multiplayer, but definitelly noticed that most players are playing as Spazz. A shame though, that by default 2 out of 3 characters are not especially "useful" with their default moveset.

No, mutators are not on by default, they are pieces of script that jj2+ reads and runs when enabled by the server with the /mutators <name> on/off. They can change the game mechanics aswell as a lot of other things. Scripts can be pretty technical, but for more information you can look on this site: https://www.jazz2online.com/jj2plus/plus-angelscript.html. If you want to know more about how to run a server and configure it with server commands, i recommend checking the jj2+ readme at https://docs.jj2.plus/.

Aside from special moves, Jazz/Lori can use their copter ears while spaz can't. This can help in some very open maps, but generally the doublejump beats it, because doublejump allows for an instant turn in the air while turning with the copter ears is more slowly.