Once upon a time, I did like Neon Genesis Evangelion in large part for simple reasons that could be applied to any series I liked. There was cool creatures and giant robots, a protagonist that was very different from the norm, interesting visuals, and intriguing psychological concepts. I still consider these factors when discussing my liking for EVA, but they have considerably diminished in their importance. Instead, my liking for EVA has more to do with how I strongly I empathized with the themes and characters of the series, and the series' effective and unflinching communication of its story.
When it comes to stories, I usually select ones that justify and demonstrate my beliefs about how things are, or how they should be. Comforting material. There are exceptions, of course, but these are usually my favorite books, such as One Flew Over the Cuckoo's Nest or The Handmaid's Tale. Perhaps I take it as a given that, while TV and movies can be dark and dramatic, it's up to the books to alter you and make you question things.
That is one of the things that makes Evangelion so remarkable, for did alter me and make me question things. This was accomplished in part by ressurrecting demons I'd thought dead, reminding me of new ones, and ones that had just grew. Yet I this experience, which could have been painful, instead translated into the same nerdy glee that I approached other fandoms with, but the psychological impact did not change.
Unsurprisingly, Shinji Ikari was the center of these feelings. While I had no analogous problems with my family, I went through a period in my life (at about his age, in fact) where I believed I was worthless, and didn't want to do anything but hide. To my shame, it took the actions of others to change me. While I hated and disdained that period in my life, and my actions during it, I somehow grew to like the fact that Shinji represented it, and then like the character himself.
Perhaps it was not that Shinji represented my pain, but that he represented a thing seldom found in fiction. When Hideaki Anno wrote him, he did not hold back. While he gave Shinji a few likable aspects, such as compassion and making the decision to help others, Shinji was almost heartwrenchingly realisitc. He was not a hero...no one in the cast was.
I appreciated this, because I don't care for heroes. I want characters that feel human, usually ones that hurt. A character can be both, but to me being a hero has always meant one worthy of emulation, and I dislike the concept. Ultimately, it's up to you to give yourself qualities that are "heroic", and I prefer not to use someone else as a template.
So EVA had no heroes. No villains either, much as I despise Gendou Ikari. I adore worlds like this, because I find the concept of heroes and villians negates the ambiguity/diversity of the real world, something I believe is one of reality's best aspects.
So yes, EVA appears "comforting" in that way. But there was more. I read Shinji's story as a coming-of-age one, albiet more tormented and ambigious than most. Through a slow, painful process, Shinji reaches some form of emotional maturity. However, there were many starts and stops along the way, and he is not entirely healed. My path to maturity is becoming as slow and painful as his, and I don't know how I'll end up. But Shinji's tale reminded me of the retreating I was doing, something I did not like to face. It remains to be seen if this was a positive thing or not.
Many other characters reflected myself, but I did not develop any personal liking for most of them, though many were no more or less deplorable than Shinji. Misato, Ritsuko, Asuka, and Gendou were all like me in some ways, in emotional maturity and the tendency to push others away by being abrasive. I often wondered if my liking for Kaworu Nagisa was because I, in my heart of hearts, wanted someone "perfect" by my side, despite my condemning of a relationship between himself and Shinji. The last idea frightens me.
All this, strangely helped change my liking for EVA into something entirely different than my liking for other series, though the level of intensity remained the same. With Evangelion, I do what fans normally do; buy stuff, cultivate an obessive enthusiasm, and communicate with others who share it (in fact, I've liked Evangelion so much that I've made efforts to get others interested in it).
I like to say I prefer stuff that brings in shades of gray, breaks with convention, and confronts expectations. But there is hardly anything truly affecting on my roster. Neon Genesis Evangelion did gave me several trappings from a dozen other stories I like, but also produced a distorted mirror in which I saw myself. That makes the series unique, and therefore my current favorite.