Yes, I do believe he is a Christian representative of a Nation.
And he is the temporal leader of a Christian Church.
He represents the unity of the Church of England.
The spiritual power is owned by the Archbishop of Canterbury.
It's very similar to the RCC, but the difference is that the Pope owns both the temporal and the spiritual power of the Church of Rome.
Matthew 28:18
18 And Jesus came and spake unto them, saying, All power is given unto me in heaven and in earth.
The Anointing represents the Spiritual Power of the Monarch. As the Pope, the British Monarch is stated to have both Spiritual Power and Temporal Power Status.
The Coronation Liturgy
"... Throughout the known history of Coronations in England, the anointing has been a constant feature. Anointing emphasises the spiritual role of the Sovereign.
The Anointing is the most sacred part of the service, using the Ampulla and Spoon, the latter dating back to either Richard I or Henry II. The Ampulla was supplied for the Coronation of King Charles II in 1661 and is based on an earlier, smaller vessel, which in turn was based on a fourteenth-century legend in which the Virgin Mary appeared to Archbishop Thomas Becket and presented him with a golden eagle and a vial of oil for anointing future Kings of England.
As was the case in 1953, this ceremony will not be visible to those watching on television (or online, nor indeed for those people in the Abbey) due to the presence of a newly-commissioned screen that will be held around the Coronation Chair. It is The King’s only moment of privacy during the Service, as he contemplates how he is called by God. Canopies such as these can be traced back to the Old Testament. In the Middle Ages it was custom for Sovereigns to travel beneath such a canopy. In this context, it is to signify the presence of God over this covenant of anointing. In this Coronation it also symbolizes the embrace, enveloping power, and presence of God during this moment...."
The coronation liturgy for HM King Charles III with commentary, commissioned and authorised by The Most Reverend & Right Honourable Justin Welby, Archbishop of Canterbury.
www.churchofengland.org