Use the Feed button. It's in the screenshot. The horse eats carrots if I remember correct. You can buy carrots from the shopkeeper Bernard. But there's no harm in just clicking it first and see what happens.
If you don't see the same menu I posted, click the tools icon at Home, then the horse icon (Animals). The menu you described (Like, Dominance) I have only ever seen in the Prison Cells, and you can't keep a horse in prison, so I am not sure what it is you are looking at.
The only other way I know to interact with your animals is at home, 'Check outside " then "Check animals", but the only option for the horse there is "Name horse".