The discord servers my friends and I use are just for shit posting and using voice among like 10 of us. If it becomes annoying we can move to the next thing. We're all millennials. We can run whatever server if needed it's not a big deal.
Teamspeak, as far as I know, doesn't have a way to solve this.
That being said, after thinking about it, I actually have done what you're talking about before - just not on Discord. When I find someone, I simply add them on Steam, PSN, or whichever account the game uses.