In school we were taught that multiplication is "repeated addition." Of course, that idea breaks down when asked to add 4 to itself -3 times. I have the same intuition that multiplication is arbitrary when dealing with units. I'll do my best to explain with examples:
If we were to take 3 meters ($m$) and multiply that by 4 times, I would simply add 3 meters 4 times and get $12 m$ total. The result is one dimensional.
Now multiplying 3 meters by 4 meters is extending the 3 meters into a perpendicular dimension to a length of 4 meters to get $12 m^2$. Addition no longer makes sense here as we are not adding 3 meters to itself 4 meters times. The result is 2 dimensional.
What about when multiplying different units? Multiplying 3 Newtons by 4 meters gives 12 Newton-Meters = 12 Joules. Furthermore, the space we are working with is defined by our definition of the units. 12 N-m seems 2 dimensional (Newton dimension and meter dimension) but 12 Joules is 1-dimensional.
Is there any established way of interpreting unit multiplication or is it pretty much arbitrary?
Multiplication of different units of measurement is never arbitrary but considered in a larger context involving adding/subtracting/powers, we talk about operating with rational expressions. In extension of fractions which we can not properly add or subtract unless they were expressed in the same denomination, the units of measurement need expressed in a coherent manner.
The apparent discrepancy between the 2-dimensional nature of Newton x meter and one-dimensional Joule comes from the wrong interpretation of Joule as one-dimensional. Imagine meter as unidimensional unit of length and meter x meter as bi-dimensional unit of area.