Is it possible for animals with fur/scales/feathers to get tattoos? If the answer is 'yes', then how?
It's possible and indeed common practice to apply a code to the body of a horse which is completely readable, even with the fur and is completely permanent. The process is called freeze marking or freeze branding and relies on liquid nitrogen. More details are available here; https://en.wikipedia.org/wiki/Freeze_brand
I doubt it would work for an animal covered in scales. The closest thing I can think of for that is tortoises with owners details painted onto their shell.