For me it's usually aspirational characters, people who are similar to me but somehow better, wittier, more calm, more elegant, etc. I tend to always like the same type of character and my friends will tell me, 'You'll love character _____' because I'm that predictable about it. I'm especially partial to dark haired Victorian ladies of that type (like Vanessa Ives) because they're my physical aspiration/ideal, too.
Severus Snape is my favorite of all time. It's a complete obsession; I'll leave the details out for your collective sanities. I wish people could figure out what type he was so I could investigate why. (I've read lots of threads on it and there does not seem to be any consensus, just the same 4-5 types repeated evenly.)
Is there an aspirational MBTI 'type' for us? Is that a thing? I could be on to a theory as to why I prefer certain characters, if so. ISTJ is the benefactor to our beneficiary according to socionics, and that's one of the types I see tossed around for Snape. But that can't be right...my mom is an ISTJ and she's awful. Sweet Jesus, don't let Snape be an ISTJ. Maybe I have to draw a line between 'dumb' ISTJ and 'intelligent' ISTJ. My mom only cares about seatbelts church luncheons, and is oblivious to the finer details of conversation, implications, life, etc. Snape would hate her. I just brain vomited all over this post, so, the end.