In Isle of Dogs, Atari finds a dog in the locked cage which he initially assumes is his dog Spots. Later we discover that it's actually a different dog called "Sport", whereas Spots was carried to the animal testing facility while still in his cage. This begs the question of how did Spots' cage manage to get back to the western part of Trash Island, get locked again and contain a different dog? Unless I missed something, this was not addressed in the film?
It's not the same cage. I don't believe there's anything more complicated here than the pack being mistaken about which cage they brought Atari to. It appears that many dogs were deposited on the island in cages, and at least one found himself unable to get out. The pack concluded it was Spots, but they were simply wrong.
Two dogs. Two tags. Two cages. Simple mix-up.