The Caspian tiger hasn't actually returned from extinction. What happened is that DNA testing revealed Caspian tigers and Siberian tigers were the same subspecies, not distinct populations. The Caspian population was wiped out by the early 1900s, and that extinction is permanent. There's no population that came back.
What you're probably hearing about is a reintroduction project in Kazakhstan that aims to restore tigers to the Caspian region using Siberian tigers (which are genetically identical to the extinct Caspians). It's a real conservation effort with some actual progress, but that's different from a species coming back on its own or being discovered alive after being declared extinct. The habitat work and breeding programs are genuinely impressive, but they're creating a new population from existing stock elsewhere, not recovering something that survived unknown.
The broader point still holds though - there are solid reasons to not give up on species we think are gone. Animals get rediscovered surprisingly often, and places we've written off can sometimes recover if given the chance. But the Caspian tiger specifically is extinct, even if we're trying to bring the subspecies back through deliberate reintroduction.