Why does amount of protons define how matter is?

Long answer: Any Chemistry textbook.

Short answer: The number of electrons of an atom is the same as the number of protons in the nucleus. This number of electrons (Identical to the position number in PSE!) defines all the chemistry of that atom.

Your day to day experience of the material world is governed by chemistry. This is at some level the science of atoms and groups of atoms.

Things like hardness, colour, toxicity and others are all largely determined by the interaction of atoms. In particular the outer coating of atoms, the electrons. Obviously the details of why element or compound A is harder or softer than compound B is incredibly diverse and complicated.

Typically the number of electrons is equal to the number of protons (but not always!) as atoms like to be electrically neutral and so for each positive proton, one negative electron will stick to the atom. So from this we can see that the number of protons indirectly affects the behaviour of an atom via the number of electrons...