Once upon a time, the Ohio River was 100% owned by the vast British "Colony of Virginia", which then spanned from the Atlantic Ocean to the Mississippi River. When West Virginia separated from Confederate Virginia in 1863 to side with the Northern States, the portion of the Ohio River forming WV's western border was given to them. Congress later passed a law that defined the river's ownership to the states to its east and south, namely West Virginia and Kentucky. So at this point, Ohio did not own any part of the Ohio River.
The U.S. Supreme Court later held that the river was owned by the states that border it and split by the low-water mark of 1793, but left it to the states to draw the final border. Finally in 1981, after years of litigation, Ohio, Indiana, and Kentucky reached an agreement. If you want to see how that boundary shifts, use Google Maps, search for "Ohio", and zoom in on the river.
