“daughters of kings.” The Davidic king had many women in his court. Traditional Christianity sees this as referring to Christian women being with Christ, but that is only assumed because the Psalm is traditionally believed to be only about Jesus, who did not have women and was not married.
“the queen.” The Davidic king would have a queen, and the couple will have sons (Ps. 45:16). Tradition says the queen is the Church, but the ancients would not have read the psalm that way, and the Church is nowhere else referred to as a queen. But when the meaning of the psalm is misapplied and is said to only be about Christ, then interpretations are invented to make what the psalm says fit theology. If the queen is the Church, then there are serious problems with Psalm 45:10, which directs her to forget her “own people” (which would be fellow Christians) and her “father’s house” (which would be the house of God).
The chronology of the psalm can be confusing at first but is understandable. Here in Psalm 45:9 the royal couple are already married, and then Psalm 45:10-14 gives details of some of what happened before the marriage. This kind of “conclusion, then details” happens all the time in life. For example, if there is a car wreck the driver of a car might say, “I was in a wreck but nobody was hurt. I was driving down the road when…,” and thus the driver starts with the finished event and then fills in details of how things happened.
The queen is said to be a woman of foreign descent, possibly from Tyre (Ps. 45:12) who was told to forget her own people and father’s house (Ps. 45:10), and she and her husband have an ivory house (Ps. 45:8). Those facts have led some commentators to suggest that this psalm is referring to the marriage of the Phoenician princess Jezebel to King Ahab, who had an ivory palace (1 Kings 22:39). But that is untenable since Ahab does not fit the characteristics of a godly king that are so prominent in the psalm. Solomon, who also married foreign women and lived in luxury, is a much more likely candidate.