Discord's "server" language is a throwback to non-centralized voice chats like Teamspeak or Ventrilo, where each group of users really would run their own chat server. I wouldn't be surprised if they drop that language eventually, especially since most of the users who would have recognized it have probably already migrated.
Interestingly, Discord's developer documentation and API still consistently calls them "Guilds", which I find a much better name, personally, but of course is even more of a term with gaming baggage, and "Server" was the eventually official neutral choice after that.
That's part of what I mean by gaming baggage, it's generally not a common word in contemporary English parlance unless you are a gamer or a history buff (and obviously there's an overlap in that Venn Diagram; part of why games use the term so often is how much medieval-inspired fantasy infuses the history of videogaming).
Though the medieval history is its own baggage as well that would likely keep Discord from wanting to use the term publicly in sales marketing: medieval guilds were predecessors to modern unions and there's at least a few potential enterprise companies that would balk at using a union term for chat groups.
