Christianity Explained

There are different levels to the terms "the priesthood" and "Church" which is all believers baptized into Christ, compared to presbyters given authority and the Holy Spirit by the apostles to lead and guide the "Church" into all truth. See God's Church on the Catholic Church being a divine institution instituted by Jesus and the Holy Spirit.

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Ephesians 5 can be interpreted and correlated to be talking about both, though many fathers comment on it being specifically the physical instituted Church lead by Peter representing Christ as the head. The Bible also suggests this with Ephesians 1-4 expressing unity of faith, doctrine, and rejecting falsehood.

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St. Jerome (†420)
Commentary on Ephesians 5:27
“The Church is called glorious, not because she has no sinners, but because she possesses the true faith and the sacraments of salvation.”
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Ambrosiaster (4th century) — explicit Eph 5 commentary
The Church is made holy by the truth she has received, not by the merit of her members, and she remains uncorrupted in what she teaches, though many within her are sinners.
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Theodoret of Cyrus (†c. 457) — on Ephesians 5
“He calls the Church spotless, not because no one sins, but because the doctrine entrusted to her is pure and undefiled.”
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Theodore of Mopsuestia (†428) — Eph 5 exegesis
The Church is holy because of the teaching she has received from Christ, which does not admit corruption, even though many who belong to her live imperfectly.
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St. Augustine (†430)  “The Church is holy in her sacraments and in her doctrine, though she bears sinners in her bosom.” 
“The Church, which Christ loves, is spotless because of Him. Though her members are sinners, she is holy because she is His Bride.”
“No one can be counted as having God for Father who does not have the Church for mother. The Church is one, founded on the apostles, and all who belong to her are united by the same faith and sacraments.”
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​St. Cyprian of Carthage (†258) — unity + purity of faith Cyprian directly connects Christ’s Bride imagery to doctrinal unity: “The spouse of Christ cannot be adulterous; she is undefiled and chaste.”
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St. Irenaeus (†c. 202) 
“The Church, though dispersed throughout the whole world, carefully preserves the faith… as if she possessed but one soul and one heart.”
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Revelation 21:2
“And I saw the holy city, new Jerusalem, coming down out of heaven from God, prepared as a bride adorned for her husband.”
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Revelation 21:9-10
“Then came one of the seven angels who had the seven bowls full of the seven last plagues and spoke to me, saying, ‘Come, I will show you the Bride, the wife of the Lamb.’” 10 And he carried me away in the Spirit to a great, high mountain, and showed me the holy city Jerusalem coming down out of heaven from God,

 

CCC 811

“The Church is ‘the Bride of Christ.’ She is ‘the holy People of God,’ the community of believers united by faith, baptism, and Eucharist, and guided by the bishops in communion with the Pope.”

CCC 813

“Christ loves the Church as His Bride and has given Himself up for her, so that He might sanctify her, cleanse her, and present her to Himself as a glorious Church, without spot or wrinkle.”