Trinitarian identity comes from their relationship. A husband becomes a father only when he has a child. His identity as a father is the result of his relationship with his child. Without the child, he would not have the identity of being a father. This analogy helps us understand Trinitarian identity. God the Father is the Father because he is the Father of the Son. God the Son is the Son because he is the Son of the Father. Who each person is comes from their relationship. The Father is not just the origin of the Trinity in some impersonal process, but is eternally the Father in a personal relationship with the Son. St. Thomas Aquinas identifies the proper title of the Son as 'Word'. Jesus is the Word of God since he is spoken by the Father. The identity of the Word is relational since any word requires a speaker.
The Holy Spirit is the love between the Father and the Son. The Holy Spirit has a divine will of his own and an identity of his own and is not just the spiritual union of the Father and the Son. St Thomas Aquinas identifies the title of the Holy Spirit as 'Love'. This obviously relational word properly designates his identity and relationship.
Each person totally gives of themselves and this giving is who and what each divine person is. Each divine person is a divine person because they totally give of themselves and that is what each person is, pure gift.
Trinitarian identity comes from their relationship. God the Father is the Father because he is the Father of the Son. God the Son is the Son because is the Son of his Father. The Father is eternally the Father. His identity comes from his giving and receiving. The Son's identity comes from his receiving and giving. The Son has the identity as 'Word' which St. Thomas Aquinas identitfies. The Holy Spirit has the identity of 'Love' as identified by St. Thomas Aquinas. Both words indicate identity and relationship, 'Word' as spoken by the Father and the 'Love' between the Father and the Son. Each person is a pure gift.