Yes, in the New Covenant (Testament), all believers are anointed and there is only One who is above everyone else-- the Lord Jesus Christ (Hebrews 1:9). Three verses in the New Covenant explain to us that we all have the Holy Spirit and nobody can say that they have secret revelation or God has spoken "exclusively" through him: 2 Corinthians 1:21, 1 John 2:27, and 1 John 2:20.
The difference between the Old and the New Covenant is that in the Old Testament, the kings and the priests are the only anointed ones and set apart for their particular functions and services (Exodus 30:30). Whereas, in the New Covenant ALL believers are anointed and not lacking access to any truth which has been revealed by God.
1 Peter 2:9, "But you are a chosen race, a royal priesthood, a dedicated nation, [God's] own purchased, special people, that you may set forth the wonderful deeds and display the virtues and perfections of Him Who called you out of darkness into His marvelous light."
This is the uniqueness of the New Covenant church: uniqueness of Christ as the eternal High Priest (Hebrews 7:24 & 26) with the uniqueness of His anointed people who are kings and priests forever (Revelation 1:6).
Now, the epistle of 1 John tells us that many will come and will try to bring you some secret teaching and calling themselves above everybody else. 1 John called them anti-Christ and so on. Today, certain denomination pastors are teaching that they are above other believers regarding themselves as elite and the only anointed ones. They are trying to elevate themselves above other believers. So these false pastors will use the 10 % tithes, the title of the anointed from the Old Testament, and the curse of the law to accomplish their evil gain (money). Not to say that all these denominations--Baptists, independent baptist, Pentecostal, Evangelical, and every one of them who have false pastors--are all out of context from the true church indicated in the New Covenant. 1 Corinthians 3:11,"For no other foundation can anyone lay than that which is [already] laid, which is Jesus Christ (the Messiah, the Anointed One)." All of these above mentioned denominations are under somebody else's name and they will have to give an account for all their deeds on earth (2 Corinthians 5:10).
An invitation is made for any serious denominationalist believer to get out from such sectarianism to be gathered together under the name of the Lord Jesus Christ who is the unique and only true church mentioned in the New Covenant. Remember, there's only one foundation (1 Corinthians 3:11). Wood, hay, and straw (in 1 Corinthians 3:12) belong to denominations. On the other hand, gold, silver, and costly stones belong to the true teaching of the New Covenant. It's up to you, dear believers, to receive the ultimate of it in order to glorify the Lord Jesus Christ before we get home.
Remember, because all believers are anointed, make it clear that we should refuse to listen to anyone claiming a special anointing that sets himself apart from the rest of us.