Formally, why is it meaningful to multiply, but not add, different units? I know a bit of dimensional analysis from physics.
Simple example, if you have
$10J\cdot m^{-1} \times x = 20J\cdot kg\cdot m^{-2}$
then you can deduce that $x=2kg\cdot m^{-1}$, the units being determined in the obvious, natural way.
You cannot add Newtons units to metre units, but you can multiply them to get a new unit of Newton-metres.
This is intuitively meaningful to me and requires no further explanation. But in the formal, axiom-and-theorem sense, what allows me to multiply arbitrary units, and what forbids me from adding different units?
 A: This is definitely an interesting thought. Nothing forbid you in any sense. Mathematics is an abstract art where you can do pretty much anything your want. The only reason you don't see units being added is because the interpretation of the result is not meaningful in any immediate way. But there is no axiomatic rule saying you can multiply units but not add them. 
A similar question students often ask if why when we multiply matrices, we do that weird way instead of just multiplying them component wise? The same answer holds; we don't do it because the result isn't meaningful as far as elementary mathematics goes. However in some cases it is (see Hadamard product). In that sense, if you can find a way to derive some meaning from adding units, then perhaps it would be useful to add different units.
There is actually an inherent difference between mathematics and how we choose to interpret that mathematics. In the field of pure maths, the concept of units is in some sense non existent; saying $5$ newtons is an interpretation of the abstract number $5$ into a meaningful real world thing, namely, the magnitude of a force. 
If you wanted you are free to say, I am now going to define the addition of units. Consider a $5N$ force and an area of $10 \ m^2$. I will define the unit sum (denoted $\oplus$) as:
$$
5\ N \oplus 10 \ m^2 = 15 \ N\oplus m^2
$$
where the units of the answer is $N\oplus m^2$. See what you can discover about such a system and whether you can derive any meaning from it. Do share your findings!
