I had similar thing happen but with not quite so much oil; some on his face and back from climbing under my bike to look at the sump-plug.
Problem is that my dog is jet-black so I didn't notice the oil until he'd "itched" himself; along the hall wall, up the stairs, lounge carpet, washing machine, kitchen units, etc,etc. All I can say is that a dog would make a pretty good paintbrush.
I used white spirit on the inanimate objects and we ended up using moisturiser (maybe E45) to loosen up the oil, followed by dog shampoo on the mutt.
Have fun !!!