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Short answer: one mount is confirmed. Every player has a Seeker, which ArenaNet calls both a connection to the spirits of Orr and a mount. Anything beyond that is not announced.
The announcement press release confirms each player has a Seeker, described as both a connection to the spirits of Orr and a mount, tied to the Vaelwarden you play. ArenaNet has not said whether the Seeker takes different forms, how you get it, or whether other mounts exist alongside it.
Riding is one mode in a larger system. The official site describes movement that "transfers your momentum between modes of travel," chaining gliding, riding, leaping, and wall running. The full picture is on the gameplay page. Momentum even feeds into combat, which rewards movement alongside skill use and positioning.
Guild Wars 2 added mounts in the Path of Fire expansion in 2017. Each has its own movement: the raptor leaps, the springer jumps high, the skimmer hovers, the griffon dives.
Guild Wars 2 mount background from the Guild Wars 2 wiki: Mount ↗, used under CC-BY-NC-SA.
Confirmed: the Seeker exists, it is a mount, and riding chains into gliding, leaping, and wall running. Not confirmed: additional mounts, flying, customization, or how the Seeker grows. Everything official lands on what we know as ArenaNet reveals it.