Symbolism, Imagery, Allegory
Like the capital-P Presence, the capital-F Feeling is rarely referred to. It's a flash of intuitive certainty that comes over Dan at certain times, and is effectively a wise Socrates within him.
Here's Dan's first encounter with the Feeling. When anxieties arise over resuming school, Dan calms himself, thinking: “Old urges continue to arise, but urges do not matter; only actions do. A warrior is as a warrior does.” Startled at his own confidence, Dan then thinks: “It was a feeling-certainty, a knowing. It was as if Socrates was inside me, a warrior within. This feeling was to remain with me” (4.392-393).
Later, the Feeling tells Dan the giant in one of his visions is his mind (4.426). In another vision, the Feeling advises him to stay within the pentacle while confronting monsters in an attic (6.89).
Perhaps the point of the Feeling is to show that those who attain such discipline as Dan's become blessed with certain (capitalized!) powers.