I do have wireguard on my server as well, I guess it’s similar to what tailscale does?

Tailscale uses wireguard but adds a coordination server to manage peers and facilitate NAT traversal (directly when possible, and via a intermediary server when it isn't).

If your NAT gateway isn't rewriting source port numbers it is sometimes possible to make wireguard punch through NAT on its own if both peers configure endpoints for eachother and turn on keepalives.

Do you know if Yggdrasil does something similar and if we exchange data directly when playing over Yggdrasil virtual IPv6 network?

From this FAQ it sounds like yggdrasil does not attempt to do any kind of NAT traversal so two hosts can only be peers if at least one of them has an open port. I don't know much about yggdrasil but from this FAQ answer it sounds like it runs over TCP (so using TCP applications means two layers of TCP) which is not going to be conducive to a good gaming experience.

Samy Kamkar's amazing pwnat tool might be of interest to you.