Yes, because Unity is running an earlier version of mono, TNet can't make use of dual mode sockets. Because of this, ipv4 servers can only accept ipv4 clients, and ipv6 servers can only accept ipv6 clients. Likewise for clients connecting to servers, but that's based more on the IP you enter rather than the public IP you've been assigned (as most ISPs are dual-stack now).
Back on topic:
if you want to connect to an ipv6 server, you must address it using its ipv6 address. Looking at the server console output, it doesn't look like your server has been assigned a public ipv6 address? That's very strange. I don't know enough about ipv6 to recognize if those local IPs are accessible over the WAN. I know the fe80:: addresses are link-only, but the 2002: one looks strange as well.
Regardless, the line that outputs the external IP should return ipv6 if you have a public ipv6. I suppose it's possible the WebClient is using the wrong interface? Still, then, one of the outputted ipv6 local addresses should be publicly accessible.
TL;DR I don't think your host gave you an ipv6 address. Double check that. If you do have a public ipv6 address and still can't connect to it, there's a bug in TNet.
edit: On your server, try going to
http://ipv6.icanhazip.com and see what it returns (it's not a fancy webpage, only a single string should pop up).