Origin of the Chinese word for "organic" for produce, etc.?

Ever since learning that the word 有機 (yǒujī) means "organic" with respect to produce, I had difficulty coming up with a satisfactory reason why it should.  The two characters just don't make any sense (well, to me, anyway) for such a meaning, and multiple native Chinese speakers have not been able to explain the origin of / rationale behind 有機 to me, either.

I finally came up with a possible reason: the French word for organic is biologique, the middle part of which bears a decent phonetic similarity to 有機.

This may not be as far-fetched as it might initially seem. The Chinese word for latte is 拿鐵 (nátiě), which is a purely phonetic conversion. Perhaps something similar occurred for "organic", from the French word for it?

Update 1:

Alternately, the definition of 機關 (jīguān) is "organ",  but that is in the sense of an organization/office, so it's not related to this meaning of organic, but perhaps whoever came up with 有機 mistakenly thought it did?

Update 2:
A native Cantonese speaker suggested that 有機 in Cantonese was chosen as being a decent phonetic match with "organic", further pointing out that Hong Kong would have been a natural place where Chinese language equivalents were created for foreign words. This idea seems to be the most reasonable explanation.