After discovering flamingos natural colour isn't red, pinkish as one naively may have assumed though, I looked into the matter more deeply. I also found out that many other animals do not produce the respective colour pigments matching then their outer appearance.
Some seem to use certain optical effects like Rayleigh Scattering or applying several different layers of filters to absorb wave lengths of light from the spectrum just emitting the intended colour.
After browsing through some more cases several questions arose, which of course cannot be answer all in detail here. However most interesting to me were:
Why does it seem so difficult for animals in contrast to plants to produce their own colour pigments?
Some colours seem to be more difficult to achieve than others. For example I found that melanin and several variation of it as colour pigments are produced quite commonly by animals, however blue rather difficult. As the following post indicates, blue colour pigments seem to be in general difficult to produce: Why are so few foods blue?