A Colpitts oscillator makes use of the common emitter configuration, and the indicated amplifier configuration is known to be a linear circuit. How come do we call the stated oscillator a nonlinear circuit? How would removal of the emitter bypass capacitor in the common emitter configuration hurt the whole oscillator?
Question: "How come do we call the stated oscillator a nonlinear circuit?"
Each oscillator is and must be - per definition - a non-linear circuit. This is because for a safe start of oscillation the closed-loop pole pair must be in the right half of the s-plane (loop gain >1). As a consequence, the oscillation amplitudes continuously are growing until a non-linear effect within the feedback loop reduces the loop gain to unity.
(In reality and due to some delay in the feedback loop, the loop gain will slightly "swing" around the nominal value of "1" - and the pole pair will move slightly between the left and the right half of the s-plane).
This non-linear effect can be caused (a) by a simple limiting effect (fixed supply voltages) or (b) by a non-linear gain characteristics of the amplifier.