There is a small workaround in Chrome to achieve your end goal.
Three dots in upper right > More tools > Create Shortcut...
You'll be presented with an option to rename the shortcut from the page default, and an option for "Open as window" which defaults to off.
This will create a shortcut on the desktop, with the favicon used as the shortcut icon.
With "Open as window" set to off, this shortcut will open as a tab - either in a current browser window, or it will launch a default browser window if one is not open. This version of the shortcut will operate no different than a standard tab in Chrome.
With "Open as window" set to on, this shortcut will open a new and specialized window of Chrome in Application Mode, where the tab bar is hidden, and some other specialized settings are applied. This version of the shortcut will operate as a separate application in Windows, appearing on the taskbar using the favicon.
Chrome achieves this by storing the favicon and some other bits of data locally in:
%AppData%\Local\Google\Chrome\User Data\Default\Web Applications
Hope that helps!