I have sliced ginger stored in a jar of honey, and another jar of a small ginger knob in vodka. Both were stored in the refrigerator. Both are about 2 to 3 years old. They do not have any off odors. However, they both have a white powdery substance that has settled to the bottom. Could it be mold? Or bacterial growth? Before I noticed the white substance, I started to mix up the honey-ginger and it mixed up milky cloudy. I'm inclined to throw it away, but if there's no chance something can grow in a gingered honey or vodka infusion, I'd be happy to keep any use. Appreciate any expert advice.
Crystallized (or Stem) ginger preserved in honey is a known thing. Shelf-life is supposed to be about 3 months for the homemade kind, and I'd say twice that (at a minimum) for store bought...Those are both pretty conservative. It should have been properly prepared/canned at the start, which would reduce the possibility of some nasty microbial infestation, but obviously your mileage may vary depending on who prepared it and how it was stored.
Honey is a known reservoir of C. botulinum endospores, but C. botulinum doesn't really grow in honey (the low water activity (.6) makes it an extremely unfriendly environment for microorganisms). Honey by itself has an indefinite shelf-life.
Vodka pickled ginger is also a known thing. Shelf-life estimates are all over the map, from three months to forever. Three months is definitely low: ginger is a tough, astringent root, and lasts three months without any particular preparation at all. Soaking it in high proof alcohol should easily multiply that.
Any alcohol of at least 80 proof in sufficient concentration should be sufficient to prohibit the growth of micro-organisms. Concentration is critical, however, and insufficient alcohol is no barrier to bacterial growth.
I'd say that 2-3 years is pushing it. And if you see anything that doesn't look normal, I'd toss it. Botulism is one of those things that you can't always detect via smell.
Storing ginger, an underground rhizome, which has a significant chance of having botulism spores present in an low-oxygen environment in honey is probably a very poor idea. The honey certainly will not be sufficiently acidic to inhibit the growth of the bacterium and the production of its toxin.
I cannot guess what the precipitate you see is, but I would certainly discard this product.
The alcohol based solution is probably safe, assuming it has been well sealed, and the alcohol content has not evaporated below the 40% level.