OK folks - we have two issues at hand that should be discussed separately.
1. Whether a species has undergone a genetic bottle neck due to a massive die off, or geographic isolation of a few individual from the rest, among other reasons. The genetic diversity is severely reduced. For example, if a pair of wild fish get trapped in a pond by drought, and breed for generations - would anybody question their (and the progenies') status of being wild?
2. Whether an individual has been in human captivity. What if you catch a wild fish, raise it for a while, then release it back to the wild? Do you consider it wild? Similarly, if you catch a wild, unrelated pair, their F1 progenies would be genetically identical to those that would have been produced in the wild. If you release them to the wild, do you consider them wild? This is where I believe people may disagree the most - human activity is involved but with no genetic consequence.
3. What if the fish have been subject to inbreeding in human captivity for generations, and then released back to the wild. I believe this is what was asked in the OP.
The genetic diversity is essentially lost forever, as mutations rarely happen in the time scale of our lives - the rate of sponteneous mutations varies a great deal but roughly in the ballpark of one percent nucleotide substitution per one million years. These released individuals might find a new nitch and thrive. A perfect example are the dingos.
Are dingso wild?