Why does 有机 as in 有机食品 mean organic?
As you may know, &quot;Organic&quot; in your case is originally a chemical term and its noun-&quot;Organ&quot; means &quot;a collection of tissues joined in structural unit to serve a common function&quot; from Wikipedia. Once the chemists incorrectly believed that &quot;organic compound&quot; could only be produced by lifes. I think this is how &quot;organic&quot; comes from &quot;organ&quot;. I don't find the exact origin of the word &quot;有机&quot; in Chinese, but I believe it originates as &quot;organic&quot; originates in English, from chemistry.
机 originally means

主发谓之机。—《说文》The thing that controls shoot is called 机. 《说文》is an old dictionary.
trigger on a bow. A structure used to shoot the arrow.

Some other meanings extend from the original one.

critical part(point) in a system
organ.  organ is a critical part in the body
mechanical structure. The critical part that controls the movement of a machine.

In recent times, after Chinese came in touch with sciences from the west, some pioneering scientists translated the terms in sciences. In general, they translated them in a liberal way. I think this is why “有机&quot; means &quot;organic&quot;, as you have &quot;organic&quot; from &quot;organ&quot; in English.