Well, if they mean organic as in “non synthetic” it makes sense. Synthetic fibers like nylon or polyester won’t break down in a compost heap, whereas organic fibers like cotton, linen and I suppose silk will. Unless a person was, as you put it, an “organomaniac” they probably wouldn’t worry if their cotton clothes were Organic in the sense of no chemicals were used in growing the cotton.
VernWinterbottom points out above that a small amount of synthetic fibers in your compost probably shouldn’t hurt.