You’re talking about the Korthia campaign… and it is character specific. The good news is though on alts if you’ve completed the korthia campaign you can get a hurry up at Bolvar that lets you bypass some of it, and go straight to the riding part.
It’s not account wide but it’s fairly quick for alts to unlock Korthia via the quest skip and thus bypass the eye of the jailer questline completely thus unlocking regular ground mounts. The exception is Venari rep which is tied to the teleporter. Each character will have to manually level up her rep to unlock the various maw flight points which is annoying AF. Thankfully however the world boss flight point is account wide and doesn’t appear to be tied to Venari. Likely because it got patched in later.
It should also be noted however that without the requisite Venari rep, you will get destroyed in the beastwarrens by the debuff. So each alt will have to unlock that, again, via Venari rep. Extremely annoying.
I’ve been working on a fresh 60 hunter recently, and yeah, no mount til Korthia, which is okay since he’s also a Worgen. And still got the eye, which I know is my own fault, I’m just taking my time on that character and doing stuff in order.
The any mount is character specific but with current gear, it should be incredibly trivial to do layer 8 twisting corridors with any class, to unlock the scavenger on any toon.
It’s always been character bound but many items you buy that matters are accountwide.
The only thing you do not get/buy account wide, is the benefits of the quests you get at certain rep levels (like the before mentioned beastwarens quest to safely go there)
Yeah some unlocks that come with ranks is accountwide. Not all iirc. Fact remain, the rep itself isn’t. You can still gain rep starting from 0 on every characters you have.