Almost any wood can be oiled, even if few are dense enough. I’ve played many pianos that had genuine ebony black keys, and they never felt greasy like they were “exuding” anything. My buddy Ronnie, who is a whiz electric guitarist, oils his axes’ fretboards. Most of the wear is borne by the metal fret, anyhow.
(1) I don't know what finish is put on ebony piano keys if any. You'll notice that I said rosewood "especially" exuded oil. And you oil your fretboard, whether it's ebony or rosewood. Most people use lemon oil, although there are some fretboard 'conditioners' that contain other products in addition to lemon oil. Many people clean and oil them each time they change strings. I do.
(2) Only the wear of the string is borne by the metal fret. The wear to the fretboard is a distinct and different wear, from string and finger. You'll go through many fret jobs before you'll see any wear on your fretboard.
Here's a relic Stratocaster fretboard (maple).
Notice the fretboard wear? Most common on Strats and Teles because of the finish put on the maple fretboards.
I'm not responsible for the fact that luthiers have been putting ebony and rosewood fretboards on guitars for 200+ years. Yeah, any wood can be oiled, but there's a reason those woods are chosen for fretboards. Ask your buddy, Ronnie, if he has a preference for a fretboard wood. He'll probably tell you rosewood or ebony. He sure as heck isn't going to tell you walnut, mahogany, tulip poplar, bubinga, wenge, or snakewood.