I think there must be some other restrictions placed on your professor's comment. Adding a pole to a general second order system makes it a third order system and the dynamics can change dramatically.
One example I can think of in which adding a pole or zero does not change the system dynamics dramatically is when the pole or zero is at much higher frequency than the pair of poles of the second order system. In this case, the dynamics only change at very short timescales. The step response, for example, would only change at the very beginning and the damping or oscillatory behavior later in the response would remain essentially the same.
At any rate, I wouldn't fret too much about making approximations of such simple systems. Once you are actually working with them in the field you will likely use numerical software to understand them, and such software has no need to make any approximations for a third order system.