What is the white sediment / residue found in ginger juice?

Ginger roots contain ~12% starch (potatoes in comparision ~18%) and the facts you described (settled down from juice after some time, white color, dense/thick texture) match the way how starch appears when it is collected for making Knödel (german dumplings made from raw potatoes). If you want to proof it you could either treat a potatoe in a similar way as the ginger or apply an iodine test to the sediment.

This white sediment is just something that happens with unfiltered ginger juice. Don't worry--it wasn't contamination, nor any unusual chemical reaction with your equipment.

This happens with fruit juice, too. Fresh fruit juice separates and needs to be shaken to get things reincorporated. If you leave citrus juice long enough, you'll get pulp floating at the top, then also get a watery layer and a cloudier layer.

What you are witnessing is the exact same thing, but with a smaller scale particle. When you allow ginger juice to sit, a white gluey texture layer will sink to the bottom with a more watery juice on top. You would need to use a very fine filter to separate out these solids, or you can just let them settle out naturally and pour the liquid off the top like you did. However, that white stuff has a lot of the good yummy gingerness in it, so I would want to make sure it got stirred into anything I made with my ginger juice. If you look at the fancy cold pressed juices at the grocer to find to ones with ginger in them, you'll notice they likely have the same white stuff on the bottom of the bottle (with instructions to shake it up before drinking!).