I definitely grow attached to characters but not usually too deeply.
That said, if I adore a character, I'm attached to them. This bond is usually infinitely memorable and unforgettable for me. I'd love to meet them in person!
Unusually; I even fall hopelessly, head over heels, in love with a character, as another character. But this is uncommon and usually temporary. I definitely fell in love with several anime guys; especially if I identified with one of the female characters. If I identified with a male character; I fell in love with a cute anime girl. Being genderfluid means I have no control over what gender of character Identify with, it's rolled with each show.
I especially bonded as Ash to Dawn, and to a lesser extent Serena to Ash in the respective seasons. Currently in this generation I'm bonding as Mallow to Ash. :3 This process is by far what dictates my shipping preferences, and the "bonding" is just my usual imaginary empathy running on high. Amusingly I even bonded as Cilan to Ash and Ash to Bianca too. Generation 1-3 saw a Misty to Brock, then Ash grow.
The longer the show, the more likely I form unseen relationships...if I actually identified with anyone in the show.
Ah! My Goddess, Shugo Chara: