Bardic Inspiration is magic. Emphasis mine in the quotes below.
From the PHB, page 8: "Without the uplifting magical support of bards and clerics, warriors might be overwhelmed by powerful foes."
Page 45, definition of a bard in the table: "An inspiring magician whose power echoes the music of creation."
Page 51, a vignette that depicts Bardic Inspiration in in-game terms: "A stern human warrior bangs his sword rhythmically against his scale mail, setting the tempo for his war chant and exhorting his companions to bravery and heroism. The magic of his song fortifies and emboldens them."
Same page, under "Music and Magic": "In the worlds of D&D, words and music are not just vibrations of air, but vocalizations with power all their own. The bard is a master of song, speech, and the magic they contain. Bards say that the multiverse was spoken into existence, that the words of the gods gave it shape, and that echoes of these primordial Words of Creation still resound throughout the cosmos. The music of bards is an attempt to snatch and harness those echoes, subtly woven into their spells and powers."
I can hear the sarcasm in your voice with the italics and extra a's in "magic," but why do you sneer at the idea that magical inspiration carries fewer implications about how the PCs relate to each other than mundane inspiration?