So then the anointing would be the Spirit Himself, rather than a gift given by Him?
Correct! The anointing is from Him; Himself, via the transforming power thru the Holy Spirit. The blessings are His results, it is a result of the anointing. It is not an anointing like a king anointing a knight either, it is a God who makes His relationship of humility in us to be accounted by Him. Which results in a blessing of real time power and supernatural blessings and yet even cost to ourselves for purposes He will reward in totality even later as well.
Hence Praising God is the ultimate end and aim of all we go thru. "Whoso offereth praise glorifieth Me." What does it matter whether you are well or ill. Whether you have money or none? It is all a matter of indifference, but one thing is not a matter of indifference, and that is that we are pleasing to the ears of God.
Paul had got back again by way of repentance into tune with God. 1 Timothy 1:13. and now He has his base as a worker in thanksgiving to Christ Jesus; his whole life has been brought into perfect relation to God.
(See 1 John 2:27 for reference.)