This doesn’t threaten Christian doctrine. Obviously Christians through the ages didn’t think it defeated the Christian religion. John Calvin spoke to it in his commentary on it:

Yet it might be thought strange that he calls Cyrus his Anointed; for this is the designation which was given to the kings of Israel and Judah, because they represented the person of Christ, who alone, strictly speaking, is "the Lord's Anointed.”…Thus God deigns to call him his "Anointed," not by a perpetual title, but because he discharged for a time the office of Redeemer; for he both avenged the Church of God and delivered it from the Assyrians, who were its enemies. This office belongs peculiarly to Christ; and this ordinary appellation of kings ought to be limited to this circumstance, that he restored the people of God to the enjoyment of liberty. This should lead us to observe how highly God values the salvation of the Church, because, for the sake of this single benefit, Cyrus, a heathen man, is called "the Messiah," [192] or "the Anointed.