Giftable Mounts?

[Jeeqbit]

But most mounts seem to have a horn, or some other item that actually loads the mount into one's inventory... and those items have none of that kind of indication.

Yes, they do. In the tooltip. You won't see it once you've used the horn and added it to your mount list, because then the horn no longer exists.

AND there is no tooltip like there is for gear.

There is. I don't understand why you would see a tooltip for gear but not for a horn.