Firefox only allows for you to pin to the list of tabs in Firefox you can create a shortcut but Firefox must be your default browser to open the shortcut in Firefox., If it is one of your favorites, then you can pin it to the Firefox icon but that is not directly on the task-bar.

I want my yahoo mail icon sitting on the task bar, not buried inside the Firefox menu. I'm allowed to have my Thunderbird icon on the task bar next to the Firefox Icon. But for some strange reason, my Yahoo email icon is hijacked and I can't put it on my task bar. There's nothing to be gained by putting my Yahoo mail icon inside the Firefox menu. Two clicks if it's on my desktop, two click if it's inside the Firefox menu.

forgottengods said

Firefox only allows for you to pin to the list of tabs in Firefox ...

You cannot create a shortcut to one website on your task-bar. Also Thunder Bird is a separate application, that is why your able to have Thunder Bird as a separate icon.

You may be able to try this and then pin it to the task-bar.

https://support.mozilla.org/en-US/kb/create-desktop-shortcut-website