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?