The naming box is a semi-physical entity, since Dave was able to cut it in half with a sword. It is also seen through Trollian viewports, and the trolls also try typing " Broad" and " Priick" into it.
The naming box exists in Alternia, where, using the Daedric Alphabet, it is able to give Alternia and Karkat names, though not before trying giving them names "Turdodor Fuckball" and "Bulgereek Nookstain". The other eleven trolls weren't shown with their naming boxes; however, the command to introduce them was still called "Enter name.".
In the post-Scratch universe, it is replaced with the Honorary Placronym which was already engraved when the kids were 13, because, once again, you would try stuffing in names like Barnstench Fartface and Barfbreath Turdsmirk.