Does there exist a notion that generalizes a black body to include other emission processes than electromagnetic ones? A black body by definition is a perfect absorber for all incident electromagnetic radiation. It isotropically emits electromagnetic radiation whose spectral distribution only depends on the temperature.
When we go to higher temperatures, other radiative-like processes will start to play a role as well, like neutrino emission, production of high energy electrons and positrons etc. 
Is there a generalized notion of a black body where these other forms of energy emission are taken into account? Do the characteristics of the emitted matter and radiation still only depend on the temperature?
 A: Hawking radiation from a black hole includes all particle species. For example, a black hole radiates gravitons, neutrinos, electrons, muons, etc. in addition to photons. This was first analyzed by Don Page in 1976. However, a black hole is not a perfect blackbody; there are frequency-dependent “graybody” factors.
Quantum field theory for any kind of particle can be done at nonzero temperatures, so a blackbody in principle radiates all particles that interact with the material composing the blackbody and can be completely absorbed. For massless fermions, or for massive bosons or fermions, the spectral formula is slightly different from Planck’s Law for electromagnetic radiation from a blackbody. The following reference discusses these cases
https://www.cambridge.org/al/files/4313/6681/8631/7705_Planck_blackbody_function.pdf
Note that the fundamental charge $e$ does not appear in Planck’s blackbody formula. In other words, blackbody radiation of massless bosons does not depend on how strongly they interact with matter! This is because for a blackbody we assume that any incident photons are completely absorbed. If the blackbody is assumed to be sufficiently “thick”, it can absorb them regardless of how strongly or weakly they interact with it.
If Planck’s Law is really for any massless boson, and not just for photons, does this imply that the Sun radiates just as much power in thermal gravitons as it does on thermal photons? No, because the Sun is not a perfect blackbody to gravitons... they mostly just go right through it rather than being absorbed. So for gravitons there would be a graybody factor involving G that greatly suppresses thermal graviton radiation.
