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Short answer: only one mount is confirmed. Each player has a Seeker, described by ArenaNet as a connection to Orr's spirits and a mount. Further mounts have not been announced.
The announcement press release confirms that every player has a Seeker, described as both a connection to Orr's spirits and a mount, linked to the Vaelwarden you play. ArenaNet has not confirmed whether Seekers have different forms, how they are obtained, or whether there are other mounts.
Riding is one part of a broader movement system. The official site describes a system that "transfers your momentum between modes of travel," linking gliding, riding, leaping, and wall running. Read the complete explanation on the gameplay page. Momentum also carries into combat, where movement is rewarded along with skill use and positioning.
Mounts arrived in Guild Wars 2 with the 2017 Path of Fire expansion. Their movement differs: raptors leap, springers jump high, skimmers hover, and griffons dive.
Guild Wars 2 mount background from the Guild Wars 2 wiki: Mount, used under CC-BY-NC-SA.
Confirmed: the Seeker is a mount, and riding connects with gliding, leaping, and wall running. Still unconfirmed: other mounts, flying, customization, or the Seeker's growth. Official details are collected on what we know as ArenaNet publishes it.