There are some brands of cat food that don't include dyes. They are often more expensive, so even though our cats like them pretty well, we usually don't get them.
The dye in cheese is
annatto. Yes, it's quite difficult to get out. There are uncolored cheeses as well, even processed cheeses, but kids often seem to prefer the colored ones, even though as far as I can tell the annatto doesn't add any particular flavor to the cheese.