Are there animals which produce their own colour pigments`?
Although no expert in this field, I think this question is interesting and I would like to put forward some tentative answers.

In a general evolutionary sense one might regard coloration as a secondary feature of living things — especially animals. Their primary concern is to obtain energy, synthesize structural components, reproduce etc. So the metabolism that evolves under this pressure is such as to allow them to do these things. Colour is not a primary concern. Pigments are complex molecules, and enzymes are only likely to evolve to produce them if there are already reasonably complex precursors available. I know little about plant biochemistry, and would welcome improvement of this answer, but it would seem to me that (1) plants have more complex or different metabolic pathways than animals, such as those for synthesizing chlorophyl, and (2) the pathways for producing pigment precursors are more widespread, perhaps because coloration affords a greater advantage to plants — for pollination etc.

There does appear to be pigment synthesis in animals — an article on pigmentation in bird feathers can be found on the website of The Cornell Lab, for example. That states that the pigments in bird feathers are based on three types of compounds: carotenoids (red or yellow), melanins (black to reddish brown and pale yellow) and porphyrins (pinks, browns, reds and greens).

Two of these are related to compounds the birds synthesize themselves from amino acids — melanins for skin pigmentation, and porphyrins for haem. However, it appears that carotenoids are not synthesized by the birds that employ them for coloured feathers, but are acquired from plants in their diet. 

Finally, one might argue that producing colours by optical effects has certain advantages over the use of pigments, namely that they are ‘tunable’ — can be modulated or turned on and off.