My point was that what the previous poster had mentioned "There are quantities that have the same dimensions, but shouldn't be compared or allow arithmetic between them" is caused by using incorrect dimensional formulae.

When you use correct dimensional formulae, most of these cases disappear, e.g. torque and energy do not have the same dimensions (the cases that remain are for quantities of the same kind that measure different things, like the average and the peak value of some quantity, for which some arithmetic operations that combine them may be meaningless). The problem is that there are plenty of textbooks with incomplete formulae, so one may need to analyze them, instead of having faith that they are correct.

Moreover, for the discrete quantities obtained by counting one must define different types depending on what is counted, i.e. one must not use the same type for counting horses and for counting boxes (or more realistically for the kind of quantities that can appear in complicated expressions, a molar quantity of some chemical substance must not have the same type as a molar quantity of a different chemical substance, or, when simulating semiconductors, the concentrations of electrons, holes, donors and acceptors must have different types).

No, that's wrong.