First of all terminology:
When physicists speak of radiation they primarily speak of electromagnetic radiation. When health physicists speak of radiation they include radiations of other types, alpha and beta and neutrons in addition to gamma and xrays. They have developed a system where radiation is given in Becquerel ignoring the particular source.
So when somebody says that one xray is equivalent to background radiation, they compare the becquerels that one gets from one xray to the equivalent becquerels one would get from the ambient surroundings. This background comes from cosmic ray muons (about 1 per cm^2 per second) to natural radioactivity of stones and materials, to gases in the atmosphere released by volcanoes etc. Natural radiation is mainly non electromagnetic, since high energy photons produced by close by radioactive decays are easily absorbed by the materials intervening, and photons coming from the cosmos are absorbed or interact high in the atmosphere ( another important point for life, to have an atmosphere).
Cosmic background radiation is another story, and is not health physics related :it is photons left over from the Big Bang and has very little energy, it is in the microwave part of the electromagnetic spectrum( mm) whereas x-rays are of higher energy, in the range of nanometers.
The comparison is valid because it is the result of painstaking studies of calibration and measurements.
Response to updated question:
Wouldn't those different types of waves have different properties?
Matter responds differently to the different wavelengths of photons, due to the increasing energy they carry which is proportional to their frequency and inversely proportional to their wavelength.
The column on the far right gives the energy of the photon. A micron wavelength is in the electron Volt range and can affect molecular distances and cohesion and living matter. Below that the interaction with matter is in bulk, not individual molecules and cells after the Ultraviolet level. The electromagnetic radiation that can affect health is ultraviolet and smaller wavelengths. The smaller the wavelength the larger the possibility of destruction of living cells which is the study of health physics: by, as the frequency increases, heating in depth,breaking of chemical bonds, ionizing, and finally destroying complete cell structures when going to MeV energies.