Mounts have been a major feature in Diablo 4 from the beginning—they're even a pre-order bonus—but leaked footage from the alpha and the endgame beta, as well as some datamined info, suggest that the requirement to unlock them was changed a few times before the action RPG recently went gold.
Maybe that's why the official post said you had to complete the main questline with one character to permanently unlock mounts for your account. Blizzard revised that paragraph after news outlets and players started asking it to clarify, but not until after Kotaku had reported on it, which led to some confusion among readers. Diablo 4 general manager Rod Fergusson and Blizzard president Mike Ybarra also clarified the situation on Twitter—Ybarra specifically noted that Kotaku "didn't get it wrong," but had just reported on an incorrect blog post.
The post now says that you need to complete a questline within the main story campaign to unlock mounts and their customization options for all your characters.
Clarifying, @Kotaku and others didn't get it wrong. They reported on the original blog, and we corrected it. Our bad and I didn't know the original blog was confusing and changed. Didn't mean to call anyone out. I'll delete the tweet in a bit once folks see this.April 27, 2023
During the beta weekends last month, some players had mount skins drop from unique dungeon enemies, like the Butcher. The text in the menus for the skins said that you need to finish the Donan's Favor quest to gain access to mounts. It's unclear when that quest appears, but in the beta there were NPC stable masters as early as Kyovashad in the Fractured Peaks.
In the upcoming "server slam" beta, you'll earn a mount trophy for reaching the level 20 cap and defeating the world boss, Ashava. Trophies will be one of many customization options for mounts in the final game when it launches on June 6. There will also be unique dismount attacks for each class if Blizzard hasn't changed its mind since this 2019 Diablo 4 BlizzCon panel.