Question

Why is an object's density expressed as a relationship between two units?

Answer 1

Probably for kind of the same reason that speed is expressed as a
relationship between two units.  You know, like miles per hour .

I guess the only reason is because no single unit has been invented
to describe density.

The rate of doing work or using energy would always be expressed
as a relationship between two units ... we would say that the rate of
work is "(so many) joules per second".  But the "watt" was invented,
so we can say "(so many) watts" instead.

So I guess you're right.  Density could be simpler to describe
if we only had a unit for it.  Then we wouldn't have to say "(so many)
grams per cubic centimeter".  We would just say "(so many) (new unit)".

Let's try it out:
 
"Uhhh, pardon me Professor . . . I've been working late in the lab,
and I believe I've identified a new substance, hitherto unknown to
the scientific community, and totally unexpected.  In its pure form,
the substance appears to be pink, it smells like butterscotch, and
its density is approximately 27.4 Brianas.  I think it's time we published
these findings ... with your name as lead investigator, of course."

I like it !