They are different. Sentinel-1 transmits a horizontal or vertical polarisation and receives horizontal or vertical. With a hybrid compact polarisation like in RCM or RISAT-1, a circular polarization (left or right) is transmitted and a horizontal or vertical polarisation can be received.

Keith Raney shows here how it fits in the hierarchy of imaging radar. Basically, Quad Pol is the best but compact pol is the next best thing especially if you are willing to sacrifice some information for wider swaths.

Are these elements the same as in covariance/coherency matrix? The option of Polarimetric matrix generation in SNAP produces covariance and coherency matrices but not the scattering matrix. Can you please tell me how to produce the scattering matrix in SNAP if it’s not the same as the others?

The scattering matix S is what you start with. I’m not really an expert in PolSAR, but for polarimetric correlation, you can derive the coherency matrix with Radar > Polarimetric > Polarimetric Matrix Generation > T matrix

Make sure that you calibrate your data first (select complex output). If you apply a polarimetric speckle filter as a second step, the T-matrix is automatically generated.

I just tested it for Radarsat-2 data and according to the value range it looks fine.

PolSARpro has more features regarding polarimetry, you might try it as well. I am not sure about the difference between quad and circular polarization handling in SNAP