Well, I keep a number of solvents on hand so that I might suit the solvent to the job. Not all greases and oils are that: some are complicated mixtures that respond best to unusual solvents. Then there is the matter of the fact that some solvents will readily dissolve some plastic parts and one needs to be careful of that.
Then there is the matter of WD-40. I do not know what is in that stuff, but if left to dry out and polymerize, it forms an exceptionally sticky goo that will gum up anything. The cure, I have found, is more WD-40 to loosen the goo and then removal of the residual goo mechanically.
If I really want to strip the oil from a part, and I am talking about what amounts to a normal petroleum-based oil, then I use TCE, or Trichloroethyene (the environmentally safe alternative to my favorite 1,1,1-trichloroethane). A clean dose of that stuff will leave an oily surface bone dry and squeaky clean, and the solvent can be reused many times.
Most greases will respond to normal non-polar solvents like naphtha, or benzene (yuk), mineral spirits, petroleum ether, etc. They can work better if one can soak the part for some time. Many soft greases are really only heavy oils or mixtures that approximate such. Some greases are soaps that act as carriers for the oil they contain, e.g., lithium automobile grease--where it is a lithium based soap carrier.
Occasionally one needs a little more polarity in the solvent and that gives the solvent some water compatibility. Things like methanol (yuk), ethanol, isopropanol, or acetone.
Acetone and methyl-ethyl ketone (MEK) are really good for some things, but can eat plastic in a hurry. One can glue plastic eyeglass frames back together using only acetone as a solvent. Nail polish remover works in a pinch, too, but then it is mostly acetone.