The Differential Emission Measure (DEM) analysis is one of the most used diagnostic tools for solar and stellar coronae. Being an inverse problem, it has limitations due to the presence of random and systematic errors. We present in theses series of papers an analysis of the robustness of the inversion in the case of AIA/SDO observations. We completely characterize the DEM inversion and its statistical properties, providing all the solutions consistent with the data along with their associated probabilities, and a test of the suitability of the assumed DEM model. While Paper I focused on isothermal conditions, we now consider multi-thermal plasmas and investigate both isothermal and multithermal solutions. We demonstrate how the ambiguity between noises and multi-thermality fundamentally limits the temperature resolution of the inversion. We show that if the observed plasma is multi-thermal, isothermal solutions tend to cluster on a constant temperature whatever the number of passbands or spectral lines. The multi-thermal solutions are also found to be biased toward near isothermal solutions around 1 MK. This is true even if the residuals support the chosen DEM model, possibly leading to erroneous conclusions on the observed plasma. We propose tools to identify and quantify the possible degeneracy of solutions, thus helping the interpretation of DEM inversion.
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