While I automatically identify with fictional characters while reading a book, I don't think I've encountered any that were very similar to me in more than a few aspects. Sometimes it's cognitive style, sometimes it's desires or beliefs, other times it's a certain biographical similarity "this is how I handled it too" or "this happened to me". Some characters I've felt particularly similar to for various reasons in the past are Cyrus Smith (of Mysterious Island), Adrian Monk, the narrator of Le Petit Prince, Cyrano De Bergerac, Charles Swann, Alyx (of Picnic on Paradise), and Martin Eden.
Homura Akemi from Puella Magi Madoka Magica.
Aisaka Taiga from Toradora.
Marceline from Adventure Time.
Touko Fukawa from Dangan Ronpa.
Sasha Braus from Attack on Titan.
Zuko from Avatar the Last Airbender.
Simon from TTGL.
Ryuko Matoi from Kill la Kill.
Eren Jaeger from Attack on Titan.
Holden Caulfield from The Catcher in the Rye.
Too many...they all possess different facets of my personality though.
I'm also pretty sure most/all of them are xFxP's of some sort(with the exception of Homura who is an INFJ).