A feature I would love is character titles. These titles could be rewards from certain dungeons, tied to achivements, for reaching certain goals, etc. I think it would be cool to go into Co-Op or multiplayer in general with Wizzbang, the Trog Trasher or something. Obviously, a lot of these titles would be common amongst all players if they were given for completing dungeons, but you could put some as chest rewards, or even challenge rewards - for example, completing X dungeon with three wizards grants title Y, etc
I like this idea too. Kinda like you get different titles as you post /get more likes here in the forums? Ahhh I hope one day to be called a "thaumaturge." But you said it yourself, the priblem is there'd be a lot of common titles amongst players, so at the moment I hope that the ability to name your characters, purchase /unlock different skins, and ofc gain high Elo rank give enough to make your team feel unique. Oh, and your deck design.
I don't think lots of common titles would necessarily be a killer for it though. Some people just play single player, so it would never affect them. And even with players having the same titles, people could still pick whichever ones they like.