Are you trying to take the horse with you? You do that through that "hammer" button located in your picture. Then you press the "Animals" button on bottom left, and you can take out your horse from there.
217 days ago(+1)
the naming screen only shows when your checking on them from the hut instead of the menu, so as wandering said here just go to this screen to take the horse.
217 days ago(+1)
I see now! This was one of the biggest issues I've had with this game, thank you so much <3