In Eph. 4:7 Christ's gift denotes a person's gift from the Holy Spirit, who is the gift from Christ. That gift equips us to minister within the body. Thus, it is a gift resulting from a gift from Christ. Because of the proximity and context, "he" in 4:8 and 4:11 refers to Christ. The Holy Spirit isn't mentioned there. However, other passages such as Rom. 12:6-8 and 1 Cor. 12:4-11 identify these gifts as also from the Holy Spirit. Paul didn't distinguish between Christ's gift and the gift of the Holy Spirit.
If you take gift of the Holy Spirit as objective (i.e. the Holy Spirit is the gift that is given) then Christ's gift is the Holy Spirit.
And I will ask the Father, and he will give you another Helper, to be with you forever, 17 even the Spirit of truth, whom the world cannot receive, because it neither sees him nor knows him. You know him, for he dwells with you and will be in you.
(John 14:16–17, ESV)
And while staying with them he ordered them not to depart from Jerusalem, but to wait for the promise of the Father, which, he said, “you heard from me; 5 for John baptized with water, but you will be baptized with the Holy Spirit not many days from now.”
(Acts 1:4–5, ESV)
However, if gift of the Holy Spirit is subjective, then the Holy Spirt gives the gift.
Having gifts that differ according to the grace given to us, let us use them: if prophecy, in proportion to our faith; 7 if service, in our serving; the one who teaches, in his teaching; 8 the one who exhorts, in his exhortation; the one who contributes, in generosity; the one who leads, with zeal; the one who does acts of mercy, with cheerfulness.
(Rom. 12:6–8, ESV)
Now there are varieties of gifts, but the same Spirit; 5 and there are varieties of service, but the same Lord; 6 and there are varieties of activities, but it is the same God who empowers them all in everyone. 7 To each is given the manifestation of the Spirit for the common good. 8 For to one is given through the Spirit the utterance of wisdom, and to another the utterance of knowledge according to the same Spirit, 9 to another faith by the same Spirit, to another gifts of healing by the one Spirit, 10 to another the working of miracles, to another prophecy, to another the ability to distinguish between spirits, to another various kinds of tongues, to another the interpretation of tongues. 11 All these are empowered by one and the same Spirit, who apportions to each one individually as he wills.
(1 Cor. 12:4–11, ESV)
"Christ gave" has Christ as the subject, which makes Christ's gift as the gift Christ is giving.
To me, though I am the very least of all the saints, this grace was given, to preach to the Gentiles the unsearchable riches of Christ, 9 and to bring to light for everyone what is the plan of the mystery hidden for ages in God, who created all things, 10 so that through the church the manifold wisdom of God might now be made known to the rulers and authorities in the heavenly places. 11 This was according to the eternal purpose that he has realized in Christ Jesus our Lord, 12 in whom we have boldness and access with confidence through our faith in him. 13 So I ask you not to lose heart over what I am suffering for you, which is your glory.
(Eph. 3:8–13, ESV)
The Grammar of Subjective and Objective Genitive
1. Subjective Genitive
The genitive substantive functions semantically as the subject of the verbal idea implicit in the head noun. This is common in the NT.
Wallace, D. B. (1996). Greek Grammar beyond the Basics: An Exegetical Syntax of the New Testament (p. 113). Grand Rapids, MI: Zondervan.
2. Objective Genitive
The genitive substantive functions semantically as the direct object of the verbal idea implicit in the head noun. This is common in the NT.
Wallace, D. B. (1996). Greek Grammar beyond the Basics: An Exegetical Syntax of the New Testament (p. 116). Grand Rapids, MI: Zondervan.