If you use the extension (not the separate app), you can add the icon by yourself.
So in terminal copy this command:
gedit ~/.local/share/applications/chrome-hangouts.desktop
And now paste this code and change the values with whatever your extension id is:
#!/usr/bin/env xdg-open
[Desktop Entry]
Version=1.0
Terminal=false
Type=Application
Name=Hangouts
Exec=/opt/google/chrome/google-chrome --profile-directory=Default --app-id=*extension_id*
Icon=*use_your_icon_path_here*
StartupWMClass=crx_*extension_id*
You can find your extension id by going to Tools>Extensions and from
here.
And this is the end result.
Edit:
The guide was for Chrome (since that's the one I use on my machine). For chromium I guess you just need to change the Exec=/opt/google/chrome/google-chrome
into Exec=/usr/bin/chromium-browser --profile-directory=Default --app-id=*extension_id*
for chromium (use another .desktop entry from the same path).
And the extension id is usually the same, but sometimes they update it and change it, the reason why just copy/pasting earlier answers wouldn't work.