This is perhaps more a religious question than an English one, but I found the topic interesting and so did some research. A good accounting of what anointment is in general, and regarding its relation to King Cyrus specifically, can be found here. To avoid link rot, I'll quote several relevant paragraphs:
Cyrus was anointed with a seven-fold anointing that made a young man the most powerful King in his season. Excitingly, this anointing is available to any believer who is willing to live for a great cause and to tap into the big vision of God for our world at this time.
The list goes on and on of ordinary people who were called and anointed by God for great significance in His Kingdom because they dared to believe in the God for whom all things are possible and volunteered to serve Him in His Kingdom purpose. God’s anointing is to do God-things. God’s glory is to glorify Him. God’s power is to do the extraordinary for His Name sake.
When God confirmed that Cyrus was willing to become a servant of the Most High, God took a hold of this young man’s right hand and took him on a journey of incredible victories, great exaltation and wealth, great significance in preserving the Jews from whom would come our Messiah and the Christian Church.
The Bible describes God’s calling and covenant with Cyrus:
"Thus says the LORD to His anointed, To Cyrus, whose right hand I have held --To subdue nations before him And loose the armor of kings, To open before him the double doors, So that the gates will not be shut: 'I will go before you And make the crooked places straight; I will break in pieces the gates of bronze And cut the bars of iron. I will give you the treasures of darkness And hidden riches of secret places, That you may know that I, the LORD, Who call you by your name, Am the God of Israel.” Isaiah 45:1-3.
The Seven-fold anointing includes:
- “To subdue nations before him” – Global Influence, kingdoms.
- “To loose the armor of kings” – Successful Spiritual Warfare.
- “To open before him the “double” doors, So that the gates will not be shut” - Double Opportunities for Double Blessings
- “To go before you and make the crooked places straight – level mountains.” Favor in spite of obstacles.
- “To break in pieces the gates of bronze” – To possess unreachable opportunities
- “To cut the bars of iron” – Power over “withholding spirits”
- “To give you the treasures of darkness & hidden riches of secret places” – How to see opportunities.
So, in short: it seems that anointment refers to someone who, in the Christian faith, is singled out by God to serve a certain purpose in God's name, and promises to honor and fulfill God's wishes.
DISCLAIMER: I gained this information from research, not personal knowledge. If someone with personal ties to the concept has differing information, please let me know and I will correct or recant this answer.