First, the electromagnetic spectrum is a true continuum. Compare it to the slider of a trombone. The spectrum of sounds that a trombone can produce is a true continuum.
By contrast, in the case of a keyboard instrument or a harp the range of the instument is by design a series of steps. The set of notes available to you is a discrete set.
So for emphasis, it's not the spectrum that has a quantum nature, the spectrum is a true continuum.
The aspect that does have a quantum nature is the luminosity.
For simplicity let's take a case of a source of light that is very close to monochromatic, for example the yellow light of a Sodium lamp. Macroscopically the light of a (Sodium) lamp certainly seems to be a continuous stream. However, if you keep reducing the luminocity (for example by sequentially placing filters between source and detector) then as you keep reducing the luminocity you will find that there comes a point where the light energy that makes it to the detector arrives in chunks. The light arrives in intermittent chunks that all have the same energy (In this example they have the same energy because the light source is monochromatic).
[Later edit, 30 minutes after first posting]
I noticed I have misread your question, I apologize for that.
Indeed the quantum nature of visible light is not exclusive to light, all of the electromagnetic spectrum is like that.
On whether light is made of particles
In propagation electromagnetic energy behaves as waves, which is why for a long time it was regarded as evident that propagating electromagnetic energy is a wave phenomenon. When electromagnetic energy interacts the particle nature manifests itself. The interaction that we measure is interaction with a detector. Whenever electromagnetic energy interacts with a detector (the detector measures it), it behaves as a particle.
So we have that it neither makes sense to regard light as inherently a wave phenomenon, nor does it make sense to regard light as inherently a particle phenomenon.