Well, the word "emo" doesn't really mean anything at all anymore because teenagers now use it to describe anyone or anything that shows any sort of emotion at all. Cry when your mom dies in a horrible accident? Dude, stop being so emo! Get depressed when you realize your life is a dead end and your only friend is the bottle? Hey man, you're emo! So all those people tossing that word around willy-nilly have rendered it meaningless. If you express emotion, particularly sadness, you are "emo" to them. It's a lame way of mocking and dismissing someone; presumably the person snickering and calling someone "emo" does so because they are a badass with a heart of black iron and you could slaughter their family in front of them and they'd make fun of it online or something like that.
That said, over the past few decades not only have action heroes become more emotionally complex, the introduction of sensitive and vulnerable heroes has increased. I'm not sure you could say it's more prevalent in anime than in other mediums, but it sure seems that way. Anime heroes, especially in giant robot shows (which I assume we can all trace back to Evangelion) do come across as more emotionally raw than, say, Superman or Iron Man. Shinji is exceptionally whiny in pretty much every situation so it's difficult not to apply the "emo" label (when you use its original meaning), but for Kira's character, I don't think it really applies.
The ironic thing about internet toughguys mocking sensitive characters by calling them "emo"(whether they deserve it or not; I'm certainly not saying there aren't plenty of nancy-boy anime characters) is that if you confronted any one of these nerds in real life and asked them to kill someone or threatened them with any sort of physical violence, the odds that they'd gun someone down in cold blood, smirk, and then post about it on the internet rather than peeing their pants and running away are pretty low.