I'm not just going to say COSTUMES, because, for me, the customization actually goes beyond that and incorporates so much more...
The entire game, from character creation - costume and powers selection, to the manner/pace in which you play, how often you manage to play, what level(s) you are/play as, how you go about using your mix of powers (it may be very different than the way others do), how you slot/enhance your character, the additional powers that you pick, what sort of content you prefer (stories, radio missions, street sweeping, Giant Monster hunting, Task/Strike Forces, Raids, Trials, solo, teaming, duo-ing, small-teams, leagues)... whether you enjoy min/max-ing, role-playing, what emotes suit your character, what stances your character should use when idle...
It was all so customizable so that your entire experience was almost tailor-made for you.
This allowed us to not only create the characters we wanted... but also to play them how we wanted to... all almost universally on a very equal playing field. Besides a very strict solo approach (mainly with the incarnate system), you really were not at a disadvantage to simply do what you preferred to.
It was my game... because it allowed itself to be my game, while also being everyone else's game.
And that provided such a great deal of attachment and investment.... and, most importantly, enjoyment.