Why is 'density' different for current?
Current is a one-dimensional quantity, while charge is a zero-dimensional one. It doesn't really make sense to talk about current per volume, for the same reason it doesn't really make sense to talk about, say, how many strands per length a stranded wire has--the current (and the strands of wire) are the same all along the length; not just the same number, but the same.
On the other hand, charge is a zero-dimensional quantity; a charge is located at a point, with finite extent in all three dimensions. So it makes sense to talk about how much charge there is per volume.
In more technical terms, you could say that if something is invariant with respect to a dimension, it's meaningless to consider its density along that dimension.