I’ve never thought referee was a very accurate description of the DM’s role. A referee’s role is to settle rules disputes between competing teams.The DM’s role is more comparable to the board in a cooperative euro-style board game than a referee. That said, it’s a purely semantic distinction and doesn’t really matter.
Though to be quite honest, when DMs insist on being called referees, it’s a bit of a red flag for me. It suggests a certain degree of attachment to an old-school play style that I find a bit concerning (the attachment to it, not the play style itself).