What is the difference between single channel separation and single microphone separation? Are they equal?
It really depends on what you mean by a channel. Normally a channel in a source separation scenario would the number of sensors detecting your unknown/partially known signal sources. In the audio domain sensors are referred to as microphones.
If this viewpoint is taken single channel separation and single microphone separation are equivalent. However depending on the domain the processing behind these algorithms is often different.
For the one channel (sensor) case in any signal domain, the assumptions of independent component analysis (ICA) do not hold (there are more signal sources than sensors and the number of sensors is not greater than one). Computation auditory scene analysis (CASA) can be used to address this case. Though note that the other answer is incorrect in saying that this is for only one signal source. CASA can be expanded to separate more than one source.
Single channel separation is not the same as single microphone separation in the case where the microphone captures more than one channel simultaneously.
However, the literature that I've seen on single microphone source separation, assumes that the microphone captures only one single channel (i.e. a mono recording).