Honing oil is a lubricant for sharpening stones. Case says its honing oil "is designed for use with sharpening stones when sharpening your knife" and lists it as "Light weight mineral oil", and Buck's how-to page lists petroleum-based honing oil, with water and water-based honing fluid, as a lubricant for natural sharpening stones.
Case words its honing oil in two documents. Case's product page lists it as "Used for sharpening knives with sharpening stones", and Case's 2018 safety data sheet for Case Honing Oil gives the application as "Lubricating oil" and names the substance "Distillates (petroleum), hydrotreated light naphthenic". The oil also comes in kits: Case's Sportsman's Honing Kit page lists a bottle beside the kit's soft and hard Arkansas stones.
Which stones each maker says can take honing oil, water or nothing is sorted stone by stone in do you need oil on a sharpening stone.
Knife oil is sold for a different job. Kershaw says Kershaw Knife Oil "is engineered to protect and enhance your knife's folding mechanism", and Buck's knife care page says to lubricate a knife after cleaning and recommends Buck's Wax Lubricant. Case points its honing oil at the stone and Kershaw points its knife oil at the mechanism; neither company's pages say whether one can stand in for the other, so that question goes to the oil's maker. Kershaw's Knife Oil page gives its application instructions. Follow the instructions for your knife and lubricant rather than treating either product name as permission to substitute it.