Well, I would say that you are confused about the quantization idea. For quantization to be taken into account, physics of particles should be examined. For a simple example, particle in a box problem given in the beginning of quantum mechanics courses. So, quantization does not mean only atom, it can be examined for many situations in the real life.
Another thing I should mention that blackbody radiation curves are specific to material temperature. For a specific temperature, you have less amount of low energy radiation, then, there is a peak at some energy level where we can say radiation in this energy range is highly dense, then, decreasing to the higher energies. For example, you can easily find the concept multi-color blackbody through internet. It is used to identify optically thick (means close to be a blackbody) accretion disk, let's say around black holes. Changing temperature of accretion disk from innermost regions to outermost regions produces a superposition of different temperature blackbody radiation.
So, after some brief introduction to clarify some basic concepts, I will explain why quantization concept is included for a blackbody. The process in a black body is continous fully absorption of incident photon and emition at some other energy. (spectral density or black body curve with your saying) Since we are talking about photons in a finite closed space (sure we are inside the black body), we can define blackbody as a quantized system and photons should obey "planck distribution". (quite an easy concept you can find it through internet) And the important point, emitted photons from blackbody can not violate planck distribution that's why you have actually a peak in spectral density.
There should be also some intiution obtained from density of modes I believe. This concept favors independency of shape, volume etc. but dimension of the system is quite effective for the result.
And a final note: if there is no such distribution for a blackbody it would be really hard to keep its temperature quite long durations, it will just become a normal body.