For a current system, PSVita all the way. Offbeat, obscure titles like Akiba's Trip, Danganronpa, One Piece Red and tons more are available on the Vita. It really is the go-to system for anime style games.
As somebody else already pointed out, Sony is your friend here. Depending on your tastes, just about anything Japanese will be found on a PlayStation system. PS3 and PS4 are region free, so you can import titles for these consoles without fear that they won't work. For PS1 and PS2 games, you would most likely need an import console. There were games based on anime properties such as Space Battleship Yamato and Neon Genesis Evangelion on those systems.
In addition, if there is a chance that a anime style game got a localization (ie US release), chances are that it was on a PlayStation system.