Yes - in the future you would just want to register for the club or purebred registry before getting that new horse. If you're already in the club then they'll stay registered.

Did it not let you just transfer them? I thought that was fixed, as I was able to transfer one of my horses today to my other account.

I ahve no clue about how to transfer a horse. The only way I have found is the private trade option. Taht's no problem I just set them up for a 1 coin sale one way and a more expensive sale back to transfer money.
I feel that there is a problem that horses are kicked out of registries due to the status of the owner. I can see why they can't be added to shows but that should be a two check requirement with both the owner and the horse in the registry not a ban of the horse from the registry due to the status of the owner. Strange practise

if you click on the edit button on a horse's page (where you change the name/discipline/etc), on the desktop version there is a button to transfer the horse on the right side. you click "Transfer" then click the box under it, then type a name you want to send to. on mobile the transfer button is just under where you input your horse's name and change the division. c:

Thanks I havn't looked in that row of buttoms due to that the rest of them are for very different purposes

Maybe I am missing something but what does the uncategorized stable tab have to do with registries? You manage your own stable tabs. I also didnt realize horses get removed from registries? 

i don't think it's that specifically, it's that you can't put a horse into "Uncategorized" once you've already assigned a tab. so one way is to send a horse to a side account and send it back to your main, and then just not assign a tab to the horse. the unregistered horses were a just side effect of using that method.

another way is to make a temporary tab and move all the horses you want to unassign into it, then delete the tab. doing that should move all the horses that were in it into unassigned.

Just putting in here that horses loosing their registration when they're sold/transfered to another account that is not a member of the club/registry is how it's intended to be and is not a bug. This was implemented awhile ago and needed to be put in place to stop it from causing issues elsewhere in the game so it's not something that'll be changed back. 

I didnt even know either of those were a thing

Yeah it's definitely a thing Maki. Happens to me frequently when I have shuffled horses from main to my alts.

I literally went to my side to verify this 😅 I register them and never go back, no need, so I never noticed. Same as moving to an uncategorized tab. Never needed to so didnt notice