Most proper adjectives are derived from proper nouns; for example the proper adjective Japanese is derived from the proper noun Japan. Some proper adjectives like Unitarian or Episcopal are not derived from proper nouns. Occasionally the reverse is true; for example the proper noun Hispanic is derived from the proper adjective Hispanic.
Sometimes, a word is written as a proper adjective to designate an ethnic group with a shared culture, heritage, or ancestry. This usage asserts the existence of a unified group with common goals. For example, in Canadian government documents, Native and Aboriginal are capitalized.
A proper adjective can become a common adjective when it takes on new, more remote meanings, such as chauvinistic. In addition, over time, a proper adjective can become a common adjective by convention, generally when the word has overshadowed its original reference, such as gargantuan, quixotic, titanic, or roman in the term roman numerals.
An adjective is not a proper adjective just because it is capitalized as part of a name or title. For example, great is not a proper adjective in Great Britain, and lost is not a proper adjective in Raiders of the Lost Ark.
For example, in the sentence "I need to blow my nose; do you have any Kleenex?", the word Kleenex is a proper noun, used to name the product being discussed. This is perfectly acceptable English usage, from a grammatical perspective. It would also be acceptable to say, "I need to blow my nose; do you have any Kleenex facial tissue?", where the word Kleenex is a proper adjective. The Kimberly-Clark Corporation (which owns the trademark Kleenex) takes care to use the word only as a proper adjective. The legal risk is that a trademark used as a noun can become genericized, in which case other businesses could legally use the word to refer to their products. This happened to the word "elevator", for example, which used to be a trademark but is now a common noun.