It appears to be a case of a lost item - if it was a lost tent or even a GPS unit left behind, I doubt few would think that there’s a legal obligation (though of course a moral one) to return the property to the one who abandoned it.
However, this appears to not be the case - he was ‘forced’ to abandon his dog in a dangerous situation, returned to safety, and notified others. If this dog had slipped down into a creek or a drainage hole, no one would think the fireman who rescues the dog now owns it. The original owner should pay for any and all costs incurred in retrieving his property and friend, and get the dog back.
And absolutely deserves every stone tossed towards him for taking his dog up on such a hike in the first place. Then again, don’t even know if that’s even legal anymore... might disturb a protected species habitat...
It’s not some hunk of inanimate property, but an animal we are talking about. The animal’s welfare in such a situation does—and should—come into play.