Harmonic detection using lock-in amplifier
No, a properly implemented lock-in amplifier essentially computes a single-bin Discrete Fourier Transform of the input, measuring energy (and phase) only at the frequency of interest.  Of course practical implementations will be slightly "leaky".

However, a more crude implementation of a synchronous detector which merely uses the reference signal to alternately multiply the detected signal by +1 or -1 would indeed have quite substantial response to odd harmonics.

To put it another way, if your lock-in amplifier multiples by sine waves in quadrature, it will have a degree of harmonic rejection.  While if you build something that multiples by a square wave (or two in quadrature) it will have harmonic susceptibility, because it is actually multiplying not only by a sine at the reference frequency, but also by the sines at all of the harmonics of the reference frequency which comprise the square wave.