Lock-in amplifiers, signal averaging and signal post-processing
My question is: what is the difference of this with respect to simply averaging the signal in an oscilloscope?


They do different things.  A lock in amplifier enables you to obtain an extremely small detection bandwidth (thousands to trillions of times narrower than the center frequency of a modulated signal), enabling extraction of a narrowband signal from broadband noise.  Averaging just increases SNR and will not help you extract one signal from another.

In addition, since a lock in amplifier is coherent, it has a 3dB sensitivity advantage over incoherent detection, so even at equal bandwidth it will have higher SNR than averaging.


  Additionally, reading on DSP Lock-in amplifiers, is it not equivalent to perform some post-processing techniques on a recorded signal?


You can implement a lock in amplifier using a two channel ADC to record both the clock and signal and then perform the mixing stages in post processing, and for many applications this is the most efficient option.  However, if you perform the mixing digitally, the ADC has to have enough dynamic range to record both the signal and then noise as well as enough bandwidth to sample both, which can be a problem if your signal is much, much weaker than the noise or your bandwidth is very high.  In that case, using an analog mixer may be a better approach.