The Mentor/Sidekick system. It just worked. You could be any level, and group with any other level, and it did a decent job of making sure it still worked out ok. It wasn't as robust as a truly dynamic system like One Tameril has, but it also came out a decade before that.
The customization was fun, it got down to the level that you could customize the shaders and color of the particle effects that your powers had. The power sets were varied, and didn't necessarily vie for parity - they were just built for fun. There were definitely FoTM builds, but the meta in the game didn't revolve around spreadsheets and optimal DPS rotations - it was just about what you liked - you could make anything work.
It had an in-game mission builder, and you could run player-created missions that were creative and fun. Neverwinter has something similar now, but CoH had it first.
Open-world missions and events that were common. Rift and GW2 take a lot of pride in their public quests, but CoH was the precursor and did a good job of them.
The lore and story - the world had a rich story, and the quests/story line lead you through it and made you a part of it. And the "sequel" - CoVillains, integrated into the same world, and you could have Villain characters that came at the same world, interacted with Hero characters, and it was awesome. It wasn't just adding "another faction" and enabling some PvP options.
The game had instancing early on (and was one of the first to do so), but honestly, the instancing was one of the weaker points of CoH - it got repetitive early on, and was nowhere near as interesting as the open world events. They were the necessary evil in CoH.
Champions Online, DCUO, and Marvel Online are all super-hero genres, and CO is closest in spirit, but none of them are entirely CoH replacements.
A lot of games have a lot of the good things CoH had, but I haven't come across anything that hits all those points in the same way. I think it's going to remain one of those special games that can't be replaced (like SWG), and if you missed it, I'm sorry, but there isn't anything else quite like CoH was.