What you are seeing in the browser isn’t directly related to the server (unfortunately).
- Sign the browser OUT of your plex.tv account (upper right).
- If you can clear the Plex tokens in the browser, do so, else open an incognito browser window after signing out
- In that incognito window, open http://ip.addr.of.syno:32400/web
Now talk to it.
If your LAN is RFC-1918 compliant (192.168.x.x, 10.x.x.x, 172.16.x.x-172.31,x.x) it will be fine. Anything else requires SSH tunnel due to being non RFC compliant for home Private Network (see Wikipedia)
I see you’re also using IPv6. If you don’t need that, it will help to turn it off but not required .