Important to note, to corroborate their claim, the poster tries to say Namecheap 'owns' the domain because they see namecheap.com in the Registrar URL line of the DNS record. That is just wrong. That's not what that means. It means whoever bought the domain got it through Namecheap.

The TLD just was released on Namecheap, so it's a 'gold rush' situation. '.quest' domains are getting snapped up left and right. My suspicion is that it was purchased shortly before the OP searched it.

Maybe a bug in the search feature, or the new purchase just hadn't propagated though the system, and the user was erroneously shown it as being available.