Based on a game by Alice Soft.

Do people obsess over anime characters?

(A2A) Ah, but of course they do.
Fiction has had this potential ever since the first fur-clad, prehistoric storyteller sat down near a sputtering campfire and awed his shivering tribe with larger-than-life tales of danger, love and death. And why should anime be an exception? Because it's "only" drawn? Nah.
See above, the human heart doesn't even need imagery for this - and it's perfectly natural, too: it just proves that you are capable of getting deeply emotionally invested with something.
It speaks of a passion waiting inside you...
which is a very precious thing. As said - fiction is always designed to be larger than life.
It emphasizes and exaggerates things, making them noticeable for us, almost spelling them out for us.
Sometimes it gets us thinking about fascinating issues we never realized before.
Sometimes it makes us realize emotional truths about ourselves and the human condition.
And sometimes, it can make us fall for a character.
Whether we want / are aware of it or not. Simply put, if a work / character does such things to you, it means that it somehow struck a chord with you.
There was something about the person (or the tale) that resonated deeply with something inside you, perhaps even something you aren't properly aware of.
And yes, for some reason, sad tales are the the most likely candidates to cause this. As for why exactly this one did...
well, sorry, you are the only person on this planet who can really find that out. Did it happen to me in the past? Absolutely.
And all I can say is: savor it. Sure, it will slowly fade over time, because humans are just made that way.
But try to remember it.
Because whatever it was that caused this, it may tell you something about yourself, about what you value, what you cherish - and perhaps even something about who you truly are.