Traditionally paints are indeed made from natural elements, e.g. lapis lazuli for blue. This is often quite expensive, and cheaper modern paints rely on chemistry to obtain their colors. They make a molecule knowing it will be red or blue or green. The color of a material is a matter of physics and chemistry, and how they absorb or reflect the light.
Maybe you have heard about white light being a mix of all colors. An object appearing green under a white light absorbs all the colors except green, which is reflected.
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