Register is simply a type of the language appropriate to a situation or circumstance. It includes characteristic vocabulary, sentence structure and level of formality or politeness. Nonfiction writings are one general register (with separate registers for e.g. popular nonfiction and academic nonfiction). Another, spoken, register is that used at home with one's family.
Yes, I would say that the names of the colours would appear in the same register (though they would likewise appear together in many registers--
brown, yellow and
red are quite average words. On the other hand, colour words like
puce, ultramarine and
umber might appear together in different registers.