It seems to me that the issue is more simple. When I add, I expect that
I am adding like things and the sum has the name units as each of the
things I am adding. When I multiply two dimensioned quantities, there is
no such claim. I don't expect the units to be the same, so the problem
goes away.
Interestly, in Newton's time, it was unexceptable to do such a
calculation, that is multiply or divide two quantities that had different
units. They had exactly the problem Ludwig raised.