You realize you can just go buy a new seat at Lowe’s or a Home Depot, right? Get a soft close one.

New seat needed. Not a stain. The finish is gone and getting bigger with every harsh chemical scrub. 2nd vote for soft close.

Ditto. New seat is an easy change. Easy since you have white! I just this week changed two of three 1955 colored (they are sunshine yellow, charcoal, and peach) Kohler toilet seats. Not simple to color match and they were silly expensive. I may buy new white toilets before doing that again then no special orders and easily available for reasonable prices at Lowe's, Home Depot ;0)

Just bought a Kohler soft close for the powder room. Looks great and installed in minutes. Fits better than the original.

Your seat is made of wood, or a wood product, and is then coated with a type of paint. The paint is gone, and what you are seeing is what the seat is made of where the paint is missing. It is a common problem with that type of seat. You might want to try a plastic soft close seat.