I think that this has been alluded to, but possibly not really delved into very deeply in this topic, but in my case it was always the ideal or the character of the "character" that I was attracted to. A secondary aspect is that I have a very strong imagination, strong enough that it can sometimes take over reality if I indulge in daydreaming too much and I can visually see the imaginary layered over reality. If I watch an anime or show, read a book or visual novel that has strong, developed characters then I can visualize myself in the story as an escapism, similar to playing video games. It's given me many great ideas for novels and artwork that aren't even fanfics, but more like fuel for the stories I love to create.
The first character that I was "attracted" to, was Nene Romanova from Bubblegum Crisis 2032 (not the redone nonsense). She was a geek, great with computers and computer games, very feminine and funny. Her character artwork was attractive, but her personality and motivations were what appealed to me the most. The episode "Scoop Chase" where she was featured, really opened her character and showed how she handled friendships and pressure. In essence, she became part of an ideal of the characteristics I valued in a woman. Would I have had sex with a pillow with her image on it? No as I prefer flesh and blood to cotton and polyester. But her character fueled my imagination and I could incorporate her personality into other characters in different, unrelated stories. She also became part of an ideal that I was looking for in someone to date at the time.
From Katawa Shoujo, Lilly has become that character that has stuck in my mind the most. While her character art is attractive in a 2D artwork way (especially in her peach blouse and tan dress), it is her personality and the way she carries herself, as described by the author, that attracts me the most to her. This is similar to the way that Nene's character stuck with me, yet Lilly is with me far more than Nene Romanova ever was. Why would Lilly's character stay in my thoughts more than Nene's? I believe that the defining reason, at least in my case is twofold, the medium of the visual novel which is perfect to create lasting attachments to the fictional characters as you've seen and my real life.
As one poster previously pointed out, the medium of the visual novel makes it easier to interject ourselves into the story and interact with the characters on a more personal level. Unlike a video game where everything is given to us either in animation or auditory with minimal reading, the medium of the Visual Novel forces the reader/player, to invest their imagination into the story far more than an RPG or FPS. I can zone out on any game I play, however I could never be able to do that with Katawa Shoujo and be able to effectively interact with the story.
I'm not ashamed to say that I cried my eyes out at Lilly's good ending and with the final third of her story being this ridiculous roller coaster of emotion, needless to say I was emotionally spent. But because of the visual novel being from a "first-person" perspective, with Hisao's character development being minimal in comparison to the female counterpoints of each story, I could interject more of my own personality into the character. Plus, the way Suriko wrote Hisao's reactions and reasoning behind his lack of action till the end, mirrored how I probably would've reacted and acted in the situations with Lilly furthering my immersion into the story. There were still a few points where I would've acted slightly different or said something slightly different than how the story was written, but for the most part everything was understandable from my point of view and I could've made that choice for the same reasons as Hisao did.
If Hisao's character had been developed far more and his decisions/responses more radical in departure than what I would do, then I wouldn't have been able to become so deeply immersed into the story as I did. His motivations and reasoning wouldn't have aligned with my own, therefore I would always be aware that this was someone else's story and I have no part in it but to read it. There have been plenty of novels where I have been able to view myself as the protagonist in the story, but only if the characters motivations and reactions were similar to my own. Perhaps it's the way my brain is wired, I don't know. What I do know, is that as I read Katawa Shoujo, I in essence became Hisao Nakai and it was no longer his story but my own.
People falling in love with a character from any medium, be it VN, VIdeo Game, TV Show or Movie, goes beyond what many of the people who play Katawa Shoujo feel. I think that each of the readers/players of Katawa Shoujo put ourselves into Hisao's shoes and play the story as though his perspective is our perspective. I'm married to a beautiful wife and have two wonderful sons all of whom I love. Many of my wife's characteristics are in the character of Lilly, not all but a good portion of them. Would I divorce my wife for a virtual character? Not in my life. What I feel for Lilly's character, allows me to see those characteristics in my wife and in essence let's me see her in a new way and appreciate her more because of how Lilly's path progressed.
I firmly believe that because each of the women from KS were written with such care, dedication and believability by their respective writers, many of us will take these characters and say, "she may not be real, but maybe there is someone out there who is similar to her". Hopefully because we walked in Hisao's (our) shoes and saw how his (our) actions could affect those women in KS, we ourselves will be more careful and mindful if we happen to find our own Lilly, Hanako, Emi, Rin or Shizune so that we ourselves with have good endings as well.