They're sort of neither. The peach emoji will render differently on iOS, Android, Windows. And I'm sure emoji-replacement packs are possible on Windows and Android (even though it's also guaranteed to be a virus).

So a peach emoji is not the same thing as the iOS peach-emoji-image. Similar to how changing my font doesn't change the actual characters.

I don't think including emojis was a great idea, but now that it's happened and people everywhere use them, emoji have become characters. I agree with your point, but it's already happened and so now there's not really any going back.