The Holy Spirit is a multi-faceted subject. You have to look carefully at the context to understand what a given reference means.
On one level, all people, believers or otherwise, have the spirit of God, because we know when people die, the spirit which powers them, goes back to God who gave it. Ecc 12:7:
“and the dust returns to the earth as it was, and the spirit returns to God who gave it.”
Believers are on a different plane to the above idea. Paul tells us in Rom 8:9:
You, however, are not in the flesh but in the Spirit, if in fact the Spirit of God dwells in you. Anyone who does not have the Spirit of Christ does not belong to him.
So believers don’t just have the spirit of God in them to ensure their continued survival, they have the mind of Jesus which means they have the spirit of God and the spirit of Christ in them (see Isaiah 40:13 cf 1 Cor. 2:16).
In Luke 11:13, I believe this is a reference to asking things of God, and he by his power or spirit, will grant this request according to his will as John says in 1 John 5:14-15:
And this is the confidence that we have toward him, that if we ask anything according to his will he hears us. And if we know that he hears us in whatever we ask, we know that we have the requests that we have asked of him.
This explanation is in harmony with the context of Luke 11.
And I tell you, ask, and it will be given to you; seek, and you will find; knock, and it will be opened to you. For everyone who asks receives, and the one who seeks finds, and to the one who knocks it will be opened. (Luke 11:9-10)
It is also in harmony with the parallel passage in Matthew 7:
If you then, who are evil, know how to give good gifts to your children, how much more will your Father who is in heaven give good things to those who ask him! (Matt 7:11)