Bright reds in feathers come from what’s in the bird’s food. Carotenoid compounds give this feather its red and orange colors. Carrots have this, think carotene. though eating carrots doesn’t make your skin orange. Over eons, birds have learned to metabolize these compounds into their feathers to create orange, red, and yellow feathers. If we could make our skin different bright colors by eating things like carrots, our species would have a lot more skin variety that the browns, russet reds, and blacks. Instead, our duller skin pigments come from melanin which happens to be the same compound that most bird feathers are colored with. Interestingly, the earlier bird species like ducks and pheasants never incorporated carotenoids into their metabolism to make brightly colored red, yellow, and orange feathers; but later in evolution, songbirds, parrots, and other birds did.
How these mushrooms get their red color, I don’t know.
Amanita Muscaria mushroom and Red-tailed Black Cockatoo feather