I really like the idea of getting your first mount almost immediately after you create the character, but you get it as a little pet. While you level your character, you also level up your mount, feeding it, petting it etc. and when you hit max level, your mount will be grown and rideable but you have to make a quest chain learning to ride it.
The part with the feeding and if not well fed loosing it, I'm not much of a enthusiast, I've played Black Desert Online and there your horse gets really slow when it gets exhausted and it's pretty annoying I must say.
Maybe we can customize our mounts, different saddles, armor, banners, bags etc. Maybe even evolve them (like Pokemon š) and make them look different and getting faster.
Also I like the Guild Wars 2 mounts, every mount has a unique ability, one jumps really high, other can hook up on a mountain, third flies, runs faster etc.