I can’t board a horse anywhere else, only at “Home”
When you are editing a horse and tap on “Boarded at”, but the list doesn’t show anything other than “Home”, that means you forgot to mark your contacts as stables.
Go back to the contacts page, edit the desired contacts and check their “Contact is a stable” box (don’t forget to save). After that, you can board horses owned by other people there.
You can now edit the horse again and the stable names should appear on the “Boarded at” list.
Tip: on the contact list page, you can tell which contacts are stables by the stable icon on the left.