These two stanzas go together. Each stanza begins with “I being graft in thee.” From that follows the nature of the relationship which now exists between the two. The first of these stanzas speaks of the particular relationships which have come into being. The poet primarily takes on the feminine role; the Lord the masculine. Hence he is sister, mother, spouse. Dove is neutral but in the allusion to Canticles, dove is feminine:

Song of Solomon 6:9 (KJV 1900)
9 My dove, my undefiled is but one;
She is the only one of her mother,
She is the choice one of her that bare her.
The daughters saw her, and blessed her;
Yea, the queens and the concubines, and they praised her.

The ESV translates “undefiled” here as “my perfect one.”

The one characteristic which is unambiguously male is “son”. But in this context, it is the diminutive position, because the Lord is “father.”

Sister is likewise from Canticles (or Song of Solomon). Before reading this it should be noted that “sister” carries the emphasis of the intense closeness of the relationship is not meant to suggest something untoward:

Song of Solomon 4:9–12 (KJV 1900)
9 Thou hast ravished my heart, my sister, my spouse;
Thou hast ravished my heart with one of thine eyes,
With one chain of thy neck.
10 How fair is thy love, my sister, my spouse!
How much better is thy love than wine!
And the smell of thine ointments than all spices!
11 Thy lips, O my spouse, drop as the honeycomb:
Honey and milk are under thy tongue;
And the smell of thy garments is like the smell of Lebanon.
12 A garden inclosed is my sister, my spouse;
A spring shut up, a fountain sealed.

As for “mother”, one may ask how the poet could be in the position of “mother” toward the Lord. The answer is from the Lord himself. When Jesus’ family heard he was in a house teaching, “his family heard of it, they went out to seize him, for they were saying, ‘He is out of his mind.’” Mark 2:20-21.

As the family pressed for admittance, the matter came to Jesus’ attention:

Mark 3:31–35 (KJV 1900)
31 There came then his brethren and his mother, and, standing without, sent unto him, calling him. 32 And the multitude sat about him, and they said unto him, Behold, thy mother and thy brethren without seek for thee. 33 And he answered them, saying, Who is my mother, or my brethren? 34 And he looked round about on them which sat about him, and said, Behold my mother and my brethren! 35 For whosoever shall do the will of God, the same is my brother, and my sister, and mother.

This also being another reference to “sister.”

As spouse:

Isaiah 54:5 (KJV 1900)
5 For thy Maker is thine husband;
The Lord of hosts is his name;
And thy Redeemer the Holy One of Israel;
The God of the whole earth shall he be called.

The most extensive discussion of marriage in the New Testament, Ephesians 5:21-33, speaks directly of human marriage and then applies the same to Christ and the church.

I being graft in Thee, there up do stand
In us relations all that mutual are.
I am Thy patient, pupil, servant, and
Thy sister, mother, dove, spouse, son, and heir.
Thou art my priest, physician, prophet, king,
Lord, brother, bridegroom, father, everything.

The relationship of prophet, priest, king are considered to be the formal offices of Christ, as set forth in the Westminster Confession.

It pleased God, in his eternal purpose, to choose and ordain the Lord Jesus, his only-begotten Son, to be the Mediator between God and man,1 the Prophet,2 Priest,3 and King;4 the Head and Saviour of his Church,5 the Heir of all things,6 and Judge of the world;7 unto whom he did, from all eternity, give a people to be his seed,8 and to be by him in time redeemed, called, justified, sanctified, and glorified.9
As for Father, there is the refrain made famous in Messiah:

Isaiah 9:6 (KJV 1900)
6 For unto us a child is born, unto us a son is given:
And the government shall be upon his shoulder:
And his name shall be called Wonderful, Counseller, The mighty God,
The everlasting Father, The Prince of Peace.

In the next stanza on relationship, Taylor says that by being brought into relationship with Christ, he is brought into all of Christ’s relationships. Being in Christ, the relationships an angel has toward Christ are now Taylor’s relationship:
“I thy relations my relations name.”

I being graft in Thee I am grafted here
Into Thy family, and kindred claim
To all in heaven, God, saints, and angels there.
I Thy relations my relations name.
Thy father’s mine, Thy God my God, and I
With saints and angels draw affinity.

The relating of my God-your God, my Father, your Father comes Jesus’s words as he takes leave of Mary Magdalene following the Resurrection:

John 20:17 (KJV 1900)
17 Jesus saith unto her, Touch me not; for I am not yet ascended to my Father: but go to my brethren, and say unto them, I ascend unto my Father, and your Father; and to my God, and your God.

And so these two stanzas work out the nature of the new relationships gained by the poet upon his union with Christ. First, there are the transformation of the relationships between himself and Christ; and then the transformation of his relationships to others, because he is in Christ.

It cannot be developed here, but at the Fall in Genesis 3, the totality of relationships between the humans and Creation have fundamentally changed for the worse. But here, in God’s Garden, by being brought into relationship in Christ, there is a complete restoration of relationship between God and human; human and all other creatures.