How should I clean my new brass peppermill?

You can actually use dish soap and water to clean the outside of a brass mill. You just need to get it dried completely afterwards. A hair dryer works wonderfully for that. You can also clean it using a cut lemon half and salt, which cuts through regular cooking oils and such. For the inside, grind uncooked rice, and follow that with grinding a small amount of pepper that you discard. That will absorb anything inside. I suspect what you are seeing is likely a mineral oil or other food safe grease/oil. But as 702551 pointed out, to be sure you would need to contact the manufacturer to find out.

The grease is probably food safe if the device was made by a reputable manufacturer. I realize that might not be reassuring though.

Contact the manufacturer for more authoritative guidance since no one here knows exactly what kind of grease it is.

Best of luck.