In every case where we find the word anointing(KJV) in the Old Testament it is from a Hebrew word that is a noun. Most of the time it is coupled with the word oil as in anointing oil. If I was reading this without knowing that the word anointing is a noun I would think it is an adjective. Is the word anointing a gerund? Is anointing oil a compound noun?
It's worth noting that in the Hebrew the phrase is (apologies for lack of rigorous transliteration) shemen hamishcha; that's a noun phrase composed of two nouns (Hebrew basically uses juxtaposition to create noun phrases). The English 'anointing oil' is similarly a noun phrase composed of two nouns; if the '-ing' is putting you off consider it the same as 'anointment oil'. But as the commenters have said, the specific status of the constituent words is fairly unimportant (here and generally).