4what the flamingo actually changes
It keeps you on the surface of water instead of swimming through it, so crossing an ocean stops involving your breath meter at all. Hold down and you sink under deliberately, let go and you rise. Come up from far enough down and you get flung clear into the air, which is the fastest way out of a deep pool and a reliable way to overshoot the ledge you were aiming for.
unlocking it
The Zoologist only stocks it once your Bestiary is a quarter complete, which means netting critters and killing at least one of most common enemies. That is a chunk of deliberate work rather than something that happens on its own, and it is the real cost of the mount.
fishing on it
Sitting on the flamingo while touching liquid gives a small fishing power bump. It is not enough to reorganize a fishing loadout around, but if you are fishing an ocean or a lake anyway, riding it costs you nothing and adds a little. It also cushions most of a fall, which matters more than the speed does in practice.