This question has been asked a few times, and the answer isn't easy. You see, Anime is not exactly a genre. It is a format.
An anime horror story is basically the same as any other horror story, but the characters have big eyes.
An anime Fantasy story is basically the same as any other fantasy story, but... you get the idea.
Now that said, there are certain tropes in anime that don't often appear in western storytelling. Including these tropes in a setting will give it more of an anime feel.
(Please note that I will not be discussing Hentai tropes.)
1) Harem Anime: Not a hentai trope! Harem anime involves a main character, usually a boy but it can also be a girl in high school, who winds up surrounded by members of the opposite sex that want the main character's affection. For some reason, he or she can't express his/her feelings for any of them and comedy ensues. Examples: Tenchi Muyo, Ouran Host Club, Uresei Yatsura, Ranma 1/2, Rosario + Vampire.
2) Human Piloted Giant Robots: Strangely this isn't a big western phenomenon. The giant robots in our cartoons are sentient and usually transform into cars and things. The typical Western Piloted Giant Robot is a big, slow, clunky thing that looks more like a chicken-legged tank. In anime, the robots are more like a super-suit of armor: a 30ft tall Iron Man. In fact, I would classify these types of stories as the super-hero genre. Our hero faces daily life and drama, but must always change into his "super-suit" to defeat the evil of the week.
3) The Futility of War: Having two nuclear bombs dropped on them has imprinted heavily in the Japanese psyche. So much that Japan, or even the entire Earth routinely gets destroyed despite the heroes attempts to save them. You can see in many anime war stories that the line between good and bad is often shifting and the longer the war drags on, the more the innocent casualties pile up. In Knights of Sidonia, not only do we see hundreds of teenagers splat against a building's surface like falling rain but they dedicate an entire episode to a single character introduction only to kill her by the episode's end. Examples: Gundam, Knights of Sidonia, Macross, Evangelion.
4) Japanese Mythology: One thing about Japanese horror stories are that they draw from Japanese mythology which has its own unique takes on ghosts and vampires.
Can Savage Worlds do these? Well, yes. Any system can, because these are not effects that require a game mechanic. These are plot themes and trappings.
So if you truly want to give your players an anime feel in your game, make sure to have plenty of anime imagery (printouts and character images). That may be all you need
