"Even if a magnet is broken into atoms, each atom shall be a complete magnet. If the atom is further broken into electrons, protons, neutrons, etc. even then each particle shall behave like a complete magnet."
You will learn at some point that magnetism in materials, i.e. magnetization, comes from the material having a coherent collection of "spins". Each particle has spin, some quantum-mechanical feature, and as a result you get "spin magnetic moments" (which has a direction), which induces the magnetization.
So breaking down the material doesn't affect the existence of "spins", which is what you ultimately need for magnetization.