God is spirit
Jesus gives the clearest starting point: God is spirit (John 4:24). Unlike us, God is not a creature with a physical body that dies and an inner soul that can be separated from it. He is eternal, invisible, and not limited by the features of human life.
That does not make God distant or impersonal. He knows, wills, loves, speaks, and acts. Scripture also says that God is not a man, so we should be careful not to imagine him as simply a larger version of ourselves (Numbers 23:19).
God is a Spirit: and they that worship him must worship him in spirit and in truth.
Why does Scripture speak of God's soul?
In a few places, God speaks of his own soul. In Isaiah's promise about his chosen servant, God says that his soul delights in him (Isaiah 42:1). This is not a technical description of God's anatomy. It tells us that God's pleasure in his servant is real and deep.
The Hebrew word often translated “soul” is nephesh. It can refer to a person's life, self, desires, or whole being, depending on the setting. So when Scripture speaks of God's soul, it is using human language to reveal God's wholehearted personal response, not teaching that God has a human-style soul.
Behold my servant, whom I uphold; mine elect, in whom my soul delighteth; I have put my spirit upon him: he shall bring forth judgment to the Gentiles.
Humans are different, but made in God's image
Human beings are made in God's image (Genesis 1:27). We are personal creatures who can know God, love, make choices, and worship. The Bible can speak of a human soul and body in ways that fit our created nature, including Jesus' warning that people may harm the body but cannot ultimately destroy the soul (Matthew 10:28).
Being made in God's image does not mean God has every creaturely feature we have. Rather, our capacity for relationship, moral awareness, thought, and love points in a limited way towards the personal God who made us.
A God who truly cares
The language about God's soul should reassure us. God is not an impersonal force. He has holy purposes and genuine delight. He is pleased with the servant described in Isaiah, a passage Christians understand as fulfilled in Jesus (Matthew 12:18).
The Spirit of God also knows the deep things of God (1 Corinthians 2:10). The Bible invites us to know God as he has revealed himself, while accepting that his being is greater than our categories. So it is fair to say that God has a personal inner life. But it is wiser not to picture him as having a soul identical to ours. Read these passages slowly, and let them draw you towards the God who is spirit and who makes himself known.