Is "density per unit volume" something?
Density usually specifies a ratio (or equivalent derivative) of the type "amount/unit dimension" where dimension is usually one of (distance, area, volume, hypervolume). Based on further examples of usage (see links in Todd Minehardt's answer) of the expression "density per unit dimension" where "dimension" might be "distance", "area", "volume" or other spatial dimension, the point with writing "density per unit volume" seems to be an attempt to be explicit about the dimension in the divisor (denominator). For instance, when writing  "density per unit dimension"  it is meant that "density=amount/dimension". Thus "density per unit volume" means "density=amount/volume". One could equally well say "volume density" or "area density" to be specific, but there might be ambiguity there as well, as one might confuse usage with expressions such as "number density" or "energy density" where the property in the numerator is being specified. Then "volume density"  might be interpreted as meaning "density of volume", even if farfetched. One could specify dimensions by placing the units in parentheses, for instance, as density (per unit volume) or density (per unit area) etc.