What gives a stone the identity it’s the structure it has. For example, a Diamond is Carbon with a tetraedric arrangement formic simple bonds with other Carbon atoms. On the other hand, while graphite being also only Carbon, it forms flat rings with hibrid bonds, making it brittle and “slippery” compared to the strenght of the diamond Diamond.
Now, the colour is given by the impurities, that is atoms of other elements trapped within the cristaline structure and/or defects in the lattice. You can have Yellow, Blue, Pink, and other colored diamonds (nope, Steven Universe wasn’t inventing those) depending on which impurities are inside the diamond’s structure.
Same goes for other gemstones. You have one main structure which gives it the name and the color depends on what other chemical elements are inside.
Fun fact: sapphires and rubies are both the mineral Corundum but one is blue and the other is red. Same mineral, different color, different name.
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