There's a distinction between phenomena and physical laws. Phenomena are what actually exists in the universe. Physical laws are what would happen given particular circumstances. Physical laws are the "If A, then B" rules, while phenomena are whether you actually have A or B.
For instance, if you have 100° C water at standard pressure that is boiling, then "you have water", "it's at 100° C", "it's at standard pressure", "it's boiling" are all phenomena. The general pattern of "When you have water at standard pressure, and you raise its temperature to 100° C, it boils" is a reflection of physical laws. If you go to another planet with no water, or where the temperature is always significantly above or significantly below 100° C, then you won't see water boiling, but that doesn't mean the physical laws are any different. "If A, then B" is true even if A and B don't happen to be present.
Relativity just says that you have the same physical laws in every inertial reference frame. It doesn't say that you have the same phenomena. The physical laws by themselves don't make any reference frame special, but the universe can. Relativity doesn't prohibit distinguished reference frames, as long as they're being distinguished by their interactions with what's in the universe, rather than the physical laws of the universe. For instance, a reference frame at rest with respect to the Earth can be distinguished from a reference frame traveling at 100 mph towards the Earth in that people at rest with respect to the latter tend to die rather quickly. This is due to the physical object of the Earth, rather than the physical laws being different in the reference frames.
CMB is a phenomenon, and this phenomenon appears different in different reference frames. The physical laws that govern how it interacts with matter, however, are the same in all inertial reference frames.