Omnipresence does not mean that God is everything and everything is God (pantheism). God fills the universe in a metaphysical sense, not a physical sense. No physical entity is God, though God sometimes chooses to manifest himself through a physical entity.
Omnipresence means more than that God is a spirit being. Other incorporeal spirits exist (angels, demons), but these are not rightly described as omnipresent. Though invisible, these entities are in some sense localized (see, e.g., Dan 9:21–23; 10:10–14).
Omnipresence does not mean that part of God is in one place and part of him is in another or that he is thinly spread about the universe. He is everywhere present in his whole being.
Omnipresence does merely mean that the evidences of God’s presence are everywhere. His being (not just his laws and operations) is everywhere.
Omnipresence does not mean that God manifests himself uniformly throughout his entire universe. For instance, he does not “dwell” on earth as he does in heaven (Matt 6:9, etc.); he is not “in” the wicked in the same sense that he is in believers (1 Cor 3:16, etc.).
Furthering the point above we note also that God does not displace anything with his being. As Storms notes, “God is where everything else is” (Grandeur of God, p. 89).