The Uncategorized Horses tab is all the horses that have not yet been put into tabs of their own. You don't need to move horses into their own tabs. In order to change horses into their own tabs, you can do so using their profile and clicking the edit (same way you change a horses discipline).

You go under Manage under Estate and Tabs and you can make tabs there. Players usually use tabs to sort their horses by various different categories like breeds, gender, discipline shows or halters, explore horses etc. Easier to keep track of horses vs having them all in one place.