" I have told students that we don't
even add forces on different objects together; should I not have?"
-Larry
You can add together whatever forces you might like. The important thing is
what interpretation you give to the result - ie., how do you use that
result?
The force F in F=ma must be the net force on the mass m.
Applied to a system of masses, F is the net force on the system, m is the
total mass of the system and a is the acceleration of the system center of
mass.
In your example of an isolated, two mass system, this F would be the (zero)
sum of the two interacting forces (of each on the other).
You may add together any mathematical entities of the same dimensions. The
important question is the meaning and usefullnes of the result.