Fledge is a rare verb meaning (1) to care for (a young bird) until it’s ready to fly, and (2) to grow the plumage necessary for flight. So the phrasal adjective full-fledged and the adjectival phrase fully fledged logically mean having just reached full development, though in practice the terms often mean just full or fully without the connotations of newness.
There are a couple of issues worth noting here. One is that, as a phrasal adjective, full-fledged usually takes a hyphen. This hyphenated form is preferred in the U.S. and Canada, while the unhyphenated fully fledged is preferred outside North America.
Second, full-fledge is a misspelling. It could make sense if fledge were a noun or an adjective, but the word is conventionally only a verb. So when full-fledged is an adjective, fledged should always be a past participle.
When medical students are handed their diplomas, they are technically full-fledged doctors. [New York Times Well blog]
Scientists believe their growth was stunted by Jupiter’s gravitational pull and never had the chance to become full-fledged planets. [AP]
He set out to change MF Global from a midsize derivatives broker to full-fledged investment bank. [Wall Street Journal]
Few photographers in Britain have reached fully fledged celeb status. [Financial Times]
Up the steep carpeted stairs the whole thing broadens out into a fully fledged restaurant at the top. [Irish Times]
He returned as a fully fledged communist who would soon be in demand to teach Marxism to the guerillas of the FARC. [Sydney Morning Herald]