Note that the lye is potassium hydroxide, not sodium. The lye's reaction with fats is what makes soap, called saponification. I suspect a similar reaction is taking place between the wood ashes and the motor oil on your garage floors.
Actually, lye is either sodium hydroxide or potassium hydroxide. I admit that I did not know that. Sodium hydroxide is the most common form and is what is found in Red Devil Lye used for drain cleaning or paint removal. Both types of lye can be used to make soaps. I used potassium hydroxide many times to neutralize hydroflouric acid during masceration to extract fossilized spores and pollen grains from rocks in Palynology classes, but never knew it was considered lye. You learn something new here all the time.
I don't believe the dry wood ashes saponified the oil on my garage floor because there was no water involved to extract potassium from the wood ash to convert it to potassium hydroxide. I think it simply soaked up the oil better than anything I ever used including washing the spots with lacquer thinner when I could buy good DuPont lacquer thinner for $9.00 for a 5 gallon bucket. Those days are over but wood ashes will always be cheap. I still mean to try it to draw oil out of wood, but it won't be on something valuable the first time. If it works out well, you guys will be the first to know.