Maybe it's outdated information but researching on my own I read that it's through CSS, something that doesn't bother me but I thought it would have a graphical interface or something like that.
It's using the custom emoji on the user's instance, then just putting the :code: for the emoji in their display name. Like on my Mastodon account my display name is entered as hybrid havoc :1m: :rm:
:1m: and :rm: are custom emojis I have setup on the instance.