I just tried out the new TS5 on Linux Ubuntu 20.04.
But after the client was started there should be an Icon at the left, where you can see a few favorite programs and them who are opened. But for Teamspeak it just shows nothing.
You can see this in the Screenshot below.
Comment=TeamSpeak is software for quality voice communication via the Internet
The above goes into $HOME/.local/share/applications/teamspeak.desktop, although that depends on your distro/preference. Another place where you can place the file is /usr/share/applications/ and /usr/local/share/applications/ for system-wide applications.
In my case I chose to use the more classic logo (teamspeak_logo.svg) but there are several you can choose from in html/client_ui/images/icons/. Another good choice is tray_not_connected.ico if you like the zoomed-in headphone variant, but it may not work as well since it’s a .ico, although you should be able to convert it to a .png by using convert tray_not_connected.ico tray_not_connected.png (included with imagemagick package).
So, while the icon is actually shown in the application manager/launcher, and everywhere else, it isn’t in the taskbar and tab menu (windows change just as @_ニャー_3 mentioned) for some reason. The recommendations above didn’t work for me either. Any other ideas? Even that it’s not a big deal, it would of course still be nice to find a solution for it, just because it would look way better with it