if I didn't know norwegian I would think the one saying this hates the face he's pointing out ^^
Ansikt = the same as Old English 'Ansien', the word means sight, countenance, or that which is seen (i.e. Someone's face). 'Ansikt' would be equivalent to 'ansight'.
Dutch and German have similar (slightly outdated) words, aangezicht/angesicht. Clearly a common Germanic root.
I recently listened to an audio book in norwegian and throughout the book they only used the word fjes, never ansikt (as far as I can remember; maybe I only noticed the use of "fjes" since we only learn "ansikt" on duolingo). Could it differ depending on who's talking, where they're from etc?
Yes, it's dialect dependent. There are also a few fixed expressions where it's more common than in general use, for example "nye fjes" as in "new people/faces" (at work, on telly, etc.).