completely developed, trained, or established
So he is referring to words that are completely established as adjectives, from his perspective.
He begins with:
There are not only the run-of-the-mil adjectives
like good, bad, and ugly, but also various
verb forms (a driving rain, a decorated cake);
words created from suffixes like -ific, -ive,-ous-
-ful,-less, and -ic, words that do double duty
as nouns and adjectives (green); both
cardinal (two) and ordinal (second) numbers; determiners
or possessive pronouns like the,those, and my;
hyphenated adjective phrases such as high-quality; and so-called
attributive nouns, such as the first word in the phrase company man, wedding cake, and motel room
Then he provides us with a definition (or at least a test):
Not all of these make the grade as full-fledged adjectives
One fairly reliable test is whether a word can be modified by an adverb--for example, very, almost, or absolutely. Colors certainly qualify
and numer are usually seen as doing so as well; we could say, "Susie is almost three." But the, those, my, company, wedding, and motel
( in the above examples) are not adjectives,
despite the fact that they modify or describe nouns.
Unlike the terms simple present and simple past in grammar, full-fledged adjective does not seem to be a universally accepted name for these kinds of adjectives. From what I gathered, an adjective is "full-fledged" if passes his test. Of course, he mentions that the test is "fairly reliable", which suggests that it can fail. He might refer to more rules in the book. But I think the best definition he could come up with is that test. You might want to reach out to him if you want fully developed definition.
Further, the word fun is a full-fledged adjective because it passes his test. I think you misunderstood what he meant. He wrote
A classic example is fun, which started out as an
attributive noun, in such phrases as fun house [...]
So, at first it was a kind of noun, but since then the word fun has "edged" it's way into "this" category, meaning the category of "full-fledged adjectives".