There are two distinct question here so let me set out the points separately.
Ps 45 - a Bridal Psalm
There is no secret that Ps 45 appears to be a bridal psalm for Solomon when he married the princess of Tyre (V12). However, that does nor prevent, but actually requires, that the author of the psalm recall who is the true king of Israel - the God of heaven as is made abundantly clear in many places. 1 Sam 8:7, 8, 24:6, 2 Sam 19:21, 1 Chron 28:5, 29:23, 2 Chron 9:8, 13:8, Ps 5:2, 44:4.
Thus, in the midst of the psalm about the marriage of Israel's earthly king, there is a reminder, in the form of a messianic prophecy of Christ, that Solomon ruled as a deputy of the true king of Israel, Jesus Christ, as confirmed by Matt 1:1, 20, 9:27, 12:23, 15:22, 20:30, 15, 21:9, 15, Mark 10:35, Luke 1:32, 33, 18:38, 39, John 1:49, Acts 13:32-37, Heb 1:8. See also Rev 11:15, 19:16. Such a Messiah was prophesied long ago: Ex 15:18, Ps 10:16, 61:7, 68:16, 92:8, 93:5, 146:10, Isa 9:7, 47:7, Lam 5:19, Micah 4:7, etc. Compare Isa 55:3 with Acts 13:34 and John 1:49.
This is further confirmed by Heb 1:9 which quotes Ps 45:7 and applies that prophecy directly to Jesus Christ as God. Thus, it was Jesus Christ who is the one anointed.
There have been at several interpretations of who the companions/fellows were as listed in Ps 45:7 and Heb 1:9
Clearly the immediate context of 'fellows" are the attendants to the bridegroom. But what does this mean in the case of Jesus Christ? We have the following credible options:
- Eph 4:8 - “When He ascended on high, He led captives away, and gave gifts to men.” That is, as Jesus triumphantly ascended to heaven to crowned at His resurrection, as per the symbolic description in Rev 5, Jesus took with Him "many captives" as "first fruits" (1 Cor 15:20, 23, Rev 14:4, etc), ie, presumably those who were raised in Matt 27:52, 53.
- Other kings of the earth which is why Jesus is called "King of kings, and Lord of lords" (Rev 17:14, 19:18)
- As a human being (among other things) Jesus was given a name above every name, Eph 2:9-11, and thus was exalted above his fellow humans, because of His supreme sacrifice.
Therefore God exalted Him to the highest place and gave Him the name
above all names, that at the name of Jesus every knee should bow, in
heaven and on earth and under the earth, and every tongue confess that
Jesus Christ is Lord, to the glory of God the Father.
- Companions in Heaven as recorded by the many hymns of praise to Jesus such as in Rev 5:9, 10, 12, 13.