How to make synthetic button

Are you talking about plastic buttons? Those would fit your question, being buttons made out of synthetic materials (versus natural materials like bone or tortoise shell). The concept of an entirely "natural" shirt button is an amusing one.

If you are referring to plastic buttons, they are made in plastic injection molding machines. The machines can make any type of part, as long as you create a mold for it. Typically most companies contract out for the injection molded pieces they need, paying to construct the tool and then for each piece. In that case, the machinery used doesn't matter to them.