Japanese input system do allow use of arbitrary UTF-8 symbols, but you have to type out the "pronunciation" for symbols, e.g. "nottoiko-ru" for ≠, "goudou" for ≡, "yueni" for ∴, "ramuda" for λ, etc. Just using "!=" is faster.
And my other point is that != is _harder_ to type in Japanese input mode because you constantly have to manage full-width vs half-width input.