no, names are entirely optional. for example I named my first ever wolf Amicus and nothing changed other than his name. also as Mirage said, you just gotta take it with you to finish it. if the quest isn't there at the time you visit the witch just wait until the opposite time.
This is the only option I get when interacting with every pet I have (including the horse); when I finish typing out the name and hitting enter it says "what do you want to do now?" As if I hadn't entered the name at all. This seems to be a reoccurring issue as I have done multiple playthroughs and still can't get past this simple task.
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.
216 days ago(+1)
I see now! This was one of the biggest issues I've had with this game, thank you so much <3