Is ginger soluble In water, fat or both?
No, by strict interpretation of your question, ginger itself is a plant. Plants are typically insoluble as they are composed of chemicals that are fat and water insoluble to a greater or lesser extent.
However, the major spice component in ginger is [6]-gingerol, this is a volatile ketone that is soluble in a range of organic solvents (oils/fats seem likely), but only very barely soluble in water. There are a range of other flavour/scent compounds found in ginger, some of which will be water soluble, some will be organic solvent soluble, that play a role in the taste of ginger, but these are too abundant to go into.
Note that water or fat solubility has only a partial influence on your ability to taste the ginger. You can make tea from ginger and taste and feel very strongly the gingerol, but this is because the gingerol is forming a fine layer of oil on the surface of the water and you can taste minute amounts of it. In addition, it scores 60,000 on the Scoville scale - similar in range to a cayenne pepper and much hotter than a jalapeno, so in a tea with nothing to cushion the effect, the heat is very obvious.