Diablo has always had multiplayer over Battle.net via UDP/IP (with port forwarding, plus free account, but without a free server[1]), IPX[2] (LAN), modem, and serial.
[1] Though some third parties created battle.net-compatible matchmaking servers, Blizzard always took a very hostile view to such things. Getting an account on the official server is free and does not require an access key, but there is no guarantee that the server will remain up (and indeed, it was down for several weeks earlier this year), and without a sanctioned free alternative, you may one day find that there is no online matchmaking available. Whether you consider the game DRM-free is a matter of definitions. You don't need to satisfy a DRM server to install the game, nor to run it in single player, nor to run multiplayer via IPX, modem, or serial. If you want to play over the Internet, you need either the official matchmaking server or an unauthorized clone server.
[2] Newer versions of Windows removed the ability to use IPX.
I have read claims that the game can play over LAN via UDP/IP if you replace some of Diablo's files with corresponding files from newer Blizzard titles. This is of course completely unsupported.
Your best approach is probably the Devilution/DevilutionX projects, which should be able to offer a better multiplayer setup.
With regard to tethering, no, you don't tether to anything. The character is saved on your computer (and it's your problem not to lose it to a hard drive failure). The only link your character has to the online service is that your character's name matches your screen name online. If you try to log in with a different screen name, Diablo will rename your character accordingly. If the server goes down, you get to keep your character and can play it via local LAN multiplayer.
Post edited December 17, 2019 by advowson