I know that TSE HD captured most of the gamers back in the day-I'm not even sure if that multiplayer is still alive or otherwise prolific-but since we don't have it here on GOG, I was wondering if at least on the classic version it would be possible to play in multi, perhaps with the help of external means such as Discord.

Yeah, it's alive. I even used the Serious Sam source port and there's still servers.

GameRanger lists Serious Sam as supported.
http://www.gameranger.com/games/