Blackbody radiation describes an electromagnetic field in thermal equilibrium. Typically, we associate this radiation with a particular source, or boundary; but the blackbody spectrum itself is a Maxwell fact, not a matter fact. Real systems have many degrees of freedom, some matter, some Maxwell. In this particular example, the Maxwell degrees of freedom have a temperature of 3K, while the matter degrees of freedom have a much higher temperature.
Clearly, then, the matter is not in equilibrium with the electromagnetic field. How can this be?
Equilibration is a process that proceeds through a large number of interactions that each exchange energy between two thermal systems (or subsystems). If energy transfer is prevented, for instance by the conservation of mass and momentum, or because no post- states of an appropriate energy are available for one subsystem to evolve into, then the systems may remain far from equilibrium relative to each other.
I’d have to do some homework to verify this, but I think a correct statement is that free massive charged particles don’t gain or lose energy in net from passing radiation; and conversely, they cannot radiate. This is guaranteed by Lorentz symmetry, btw: if you examine a million-kelvin electron in a frame of reference moving alongside it, you’d agree there’s no way it can be radiating: there’s a frame in which it is stationary, and stationary charges don’t radiate. If it isn’t radiating in one inertial frame, it isn’t radiating in any inertial frame.
Only during collisions between charged particles is there a possibility for the matter and Maxwell degrees of freedom to exchange energy: and these are very rare in tenuous plasma. (We would call these “three-body interactions” because the photon counts as one.)
[edit to add]
Now one might object that charged particles are always interacting, because their fields fall off like 1/r^2; they don’t just end. Indeed! But one of the effects of being in a plasma is that charges are “screened”. You can think of that as meaning the net effect of many-body interactions can be described as altering the “effective” field of each particle to fall off much faster, like e^-r.