That's not saying much.
Dolphins are/were common pretty much globally, and dolphins are commonly seen in all Australian waters. Literally any costal city in Australia, and 99% of the world, could feasibly carry a dolphins brand using the standards you are laying out.
The dolphin on Redcliffe's badge looks like a Bottlenose as well, which is the most common species of dolphin in the world and the same species as Miami's mascot.
In other words, they're using literally the most generic dolphin in the world instead of a rarer endemic species like the Australian humpback or snubfin that are only found in Australian, New Guinea, and probably Indonesian waters. Though TBF I don't think that Snubfin's have been found anywhere near as far south as Moreton Bay.
Same as above basically. Eagles are found on every continent except Antarctica.
Though Manly's badge looks like a bald eagle (definitely not native to Australia lol), it's meant to be a white-bellied sea eagle which is found across India, China, South East Asia, the Pacific, and Australia. Just like with Dolphins literally any costal city, and a lot of inland cities as well (basically anywhere there're large bodies of water), could feasibly carry the white-bellied sea eagle as their mascot.
BTW; though Brumbies are commonly associated with the Snowy Mountains for cultural reasons, the largest populations of feral horses in Australia actually reside in the tropics, Qld and the NT to be specific, and are common in SEQ as well. They were common in Brisbane within living memory as well, though urban development has more or less destroyed all their habitat within the city and pushed them further and further out.
In other words, using your standards, a wild horse mascot works as well for Brisbane as dolphins, sea-eagles, or any other relatively common species would.