The expression (referring to the LORD, YHWH) is common in the OT:
- Ex 15:2, The LORD is my strength and my song, and He has become my salvation. He is my God, and I will praise Him, my father’s God, and I will exalt Him.
- 1 chron 29:3, Moreover, because of my delight in the house of my God, I now give for it my personal treasures of gold and silver, over and above all that I have provided for this holy temple:
- Ezra 9:6, saying: “O my God, I am ashamed and blush to lift my face to you, my God, for our iniquities have risen higher than our heads, and our guilt has mounted up to the heavens.
- Neh 13:14, Remember me for this, O my God, and do not blot out my deeds of loving devotion for the house of my God and for its services.
- Ps 15:1, My God, my God, why have you forsaken me? Why are you so far from saving me, so far from my cries of anguish?
- Ps 42:11, Why are you downcast, O my soul? Why the unease within me? Put your hope in God, for I will yet praise Him, my Savior and my God.
- Ps 43:4, 5, Then I will go to the altar of God, to God, my greatest joy. I will praise You with the harp, O God, my God.Why are you downcast, O my soul? Why the unease within me? Put your hope in God, for I will yet praise Him, my Savior and my God.
- Ps 59:10, My God of loving devotion will come to meet me; God will let me stare down my foes.
- Ps 63:1, You, God, are my God, earnestly I seek you; I thirst for you, my whole being longs for you, in a dry and parched land where there is no water.
- Ps 68:24, They have seen Your procession, O God—the march of my God and King into the sanctuary.
- Ps 118:28, You are my God, and I will give You thanks. You are my God, and I will exalt You.
- Micah 7:7, But as for me, I will look to the LORD; I will wait for the God of my salvation. My God will hear me.
- Hab 1:12, Are You not from everlasting, O LORD, my God, my Holy One? We will not die. O LORD, You have appointed them to execute judgment; O Rock, You have established them for correction.
… and so forth - there are many more.
The point is this, in every instance of this occurring, "my God" (referring to the LORD) is spoken by a devoted follower of the LORD. This is the distinguishing characteristic of God's faithful people - they can call Him, "my God" in contradistinction to serving any of the numerous false gods.
The reverse also occurs: God often visits people and says something like, "I am the God of Abraham, Isaac and Israel, … ". That is, these people (Abraham, Isaac and Israel) called my "my God" and therefore, did not serve false gods.
In the time of Hosea's prophecy, many people were turning away from the LORD until the LORD rejected the northern kingdom and it went into Assyrian captivity.
The same problem would occur with the southern kingdom of Judah that caused Jesus to lament: "O Jerusalem, Jerusalem, who kills the prophets and stones those sent to her, how often I have longed to gather your children together, as a hen gathers her chicks under her wings, but you were unwilling! Look, your house is left to you desolate." Here Jesus predicts that a short time later, the Jews would be rejected and the Gospel went to the gentiles instead.
Hosea 2:23 predicts the same thing:
And I will sow her as My own in the land, and I will have compassion
on ‘No Compassion.’ I will say to those called ‘Not My People,’ ‘You
are My people,’ and they will say, ‘You are my God.’”
This text is interpreted by Peter and Paul in Rom 9:25 and 1 Peter 2:10 in support of the same idea. Christians now call the LORD, "My God" along with anyone else who makes such a truthful claim, compose the people of God.